tag:blogger.com,1999:blog-74226858272753588942024-02-20T02:30:00.638-08:00Άλγεβρα - AlgebraGrΣυλλογή και εμφάνιση ενδιαφερόντων θεμάτων για την άλγεβρα και μόνο!sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-7422685827275358894.post-11409852946673404582019-04-24T02:58:00.001-07:002019-04-24T02:58:41.756-07:00(14) Άνισότητα με συνθήκη(όχι και τόσο συνηθισμένη) - mathematica.gr<a href="https://mathematica.gr/forum/viewtopic.php?f=27&t=64335">(14) Άνισότητα με συνθήκη(όχι και τόσο συνηθισμένη) - mathematica.gr</a><br /><br />
<br /><br />
Έστω $\displaystyle x,y$ δύο θετικοί αριθμοί τέτοιοι ώστε να ισχύει $\displaystyle {x^2} + {y^2} + x \ge {x^4} + {y^4} + {x^3}$.<br /><br />
Να αποδείξετε ότι:<br /><br />
$\displaystyle \frac{{1 - {x^4}}}{{{x^2}}} \ge \frac{{{y^2} - 1}}{y}$sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-61391392872649283982019-01-03T01:25:00.001-08:002019-01-03T01:25:17.955-08:00What is \(0^0\), and who decides, and why does it matter? Definitions in mathematics. | On Teaching and Learning Mathematics<a href="https://blogs.ams.org/matheducation/2018/11/01/what-is-00-and-who-decides-and-why-does-it-matter-definitions-in-mathematics/">What is \(0^0\), and who decides, and why does it matter? Definitions in mathematics. | On Teaching and Learning Mathematics</a>: By Art Duval, Contributing Editor, University of Texas at El Paso How is \(0^0\) defined? On one hand, we say \(x^0 = 1\) for all positive \(x\); on the other hand, we say \(0^y = 0\) for all posit…sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-64576057127600318042018-02-11T12:50:00.001-08:002018-02-11T12:50:31.901-08:00Real Algebraic and Analytic Geometry - Preprint Server<a href="http://www.maths.manchester.ac.uk/raag/">Real Algebraic and Analytic Geometry - Preprint Server</a>sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-14983121671116613392018-01-10T12:06:00.001-08:002018-01-10T12:06:40.797-08:00Πραγματικές ρίζες πολυωνύμου - Θεώρημα Sturm<div dir="ltr" style="text-align: left;" trbidi="on">
<img alt="" height="436" src="data:image/png;base64,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" width="640" /></div>
sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-40685409096714628652017-06-03T06:16:00.001-07:002017-06-03T06:16:03.062-07:00Change of variables - Wikipedia<a href="https://en.wikipedia.org/wiki/Change_of_variables">Change of variables - Wikipedia</a><br /><br />
<br /><br />
In mathematics, a <b>change of variables</b> is a basic technique used to simplify problems in which the original <a href="https://en.wikipedia.org/wiki/Variable_%28mathematics%29" title="Variable (mathematics)">variables</a> are replaced with <a href="https://en.wikipedia.org/wiki/Function_%28mathematics%29" title="Function (mathematics)">functions</a><br />
of other variables. The intent is that when expressed in new variables,<br />
the problem may become simpler, or equivalent to a better understood <br />
problem.<br /><br />
<br />
Change of variables is an operation that is related with <a href="https://en.wikipedia.org/wiki/Substitution_%28algebra%29" title="Substitution (algebra)">substitution</a>. However these are different operations, as it can be seen when considering <a href="https://en.wikipedia.org/wiki/Derivative" title="Derivative">differentiation</a> (<a href="https://en.wikipedia.org/wiki/Chain_rule" title="Chain rule">chain rule</a>) or <a href="https://en.wikipedia.org/wiki/Integral" title="Integral">integration</a> (<a href="https://en.wikipedia.org/wiki/Integration_by_substitution" title="Integration by substitution">integration by substitution</a>).<br /><br />
<br />
A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth order polynomial:<br /><br />
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow>
<mn>6</mn>
</mrow>
</msup>
<mo>−</mo>
<mn>9</mn>
<msup>
<mi>x</mi>
<mrow>
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>8</mn>
<mo>=</mo>
<mn>0.</mn>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{6}-9x^{3}+8=0.\,}</annotation>
</semantics>
</math></span><img alt="x^{6}-9x^{3}+8=0.\," class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f0cc55fa5f54cb9a547d58415e74c742af8e9a" style="height: 2.843ex; vertical-align: -0.505ex; width: 18.177ex;" /></span></dd></dl>Sixth order polynomial equations are generally impossible to solve in terms of radicals (see <a href="https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a>). This particular equation, however, may be written<br /><br />
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow>
<mn>3</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mn>2</mn>
</mrow>
</msup>
<mo>−</mo>
<mn>9</mn>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow>
<mn>3</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>8</mn>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x^{3})^{2}-9(x^{3})+8=0}</annotation>
</semantics>
</math></span><img alt="(x^{3})^{2}-9(x^{3})+8=0" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09319ed1f02327bb29901c90e2f39743d360a68d" style="height: 3.176ex; vertical-align: -0.838ex; width: 21.855ex;" /></span></dd></dl>(this is a simple case of a <a href="https://en.wikipedia.org/wiki/Polynomial_decomposition" title="Polynomial decomposition">polynomial decomposition</a>). Thus the equation may be simplified by defining a new variable <i>x</i><sup>3</sup> = <i>u</i>. Substituting <i>x</i> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><br />
<semantics><br />
<mrow><br />
<mstyle displaystyle="true" scriptlevel="0"><br />
<mrow><br />
<mroot><br />
<mi>u</mi><br />
<mrow><br />
<mn>3</mn><br />
</mrow><br />
</mroot><br />
</mrow><br />
</mstyle><br />
</mrow><br />
<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{u}}}</annotation><br />
</semantics><br />
</math></span><img alt="{\sqrt[ {3}]{u}}" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3256eaa81dbd56ea482d9ce5163ed7405bd1ca" style="height: 3.009ex; vertical-align: -1.005ex; width: 3.286ex;" /></span> into the polynomial gives<br /><br />
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>u</mi>
<mrow>
<mn>2</mn>
</mrow>
</msup>
<mo>−</mo>
<mn>9</mn>
<mi>u</mi>
<mo>+</mo>
<mn>8</mn>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u^{2}-9u+8=0,}</annotation>
</semantics>
</math></span><img alt="u^{2}-9u+8=0," class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f82dae9d830e35fb5f3f0ec435c53b694741379" style="height: 3.009ex; vertical-align: -0.671ex; width: 16.729ex;" /></span></dd></dl>which is just a <a href="https://en.wikipedia.org/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a> with solutions:<br /><br />
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo>=</mo>
<mn>1</mn>
<mspace width="1em"></mspace>
<mrow>
<mtext>and</mtext>
</mrow>
<mspace width="1em"></mspace>
<mi>u</mi>
<mo>=</mo>
<mn>8.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u=1\quad {\text{and}}\quad u=8.}</annotation>
</semantics>
</math></span><img alt="u=1\quad {\text{and}}\quad u=8." class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72b9693d2c7df4975f4babb5d342e904b320264" style="height: 2.176ex; vertical-align: -0.338ex; width: 20.325ex;" /></span></dd></dl>The solutions in terms of the original variable are obtained by substituting <i>x</i><sup>3</sup> back in for <i>u</i>:<br /><br />
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow>
<mn>3</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
<mspace width="1em"></mspace>
<mrow>
<mtext>and</mtext>
</mrow>
<mspace width="1em"></mspace>
<msup>
<mi>x</mi>
<mrow>
<mn>3</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>8.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.}</annotation>
</semantics>
</math></span><img alt="x^{3}=1\quad {\text{and}}\quad x^{3}=8." class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174e20b5c457fe031cfda1202b76bc64a442d350" style="height: 2.676ex; vertical-align: -0.338ex; width: 22.448ex;" /></span></dd></dl>Then, assuming that <i>x</i> is <a href="https://en.wikipedia.org/wiki/Real_number" title="Real number">real</a>,<br /><br />
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mn>1</mn>
<mrow>
<mo>/</mo>
</mrow>
<mn>3</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
<mspace width="1em"></mspace>
<mrow>
<mtext>and</mtext>
</mrow>
<mspace width="1em"></mspace>
<mi>x</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>8</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mn>1</mn>
<mrow>
<mo>/</mo>
</mrow>
<mn>3</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>2.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=(1)^{1/3}=1\quad {\text{and}}\quad x=(8)^{1/3}=2.}</annotation>
</semantics>
</math></span><img alt="x=(1)^{{1/3}}=1\quad {\text{and}}\quad x=(8)^{{1/3}}=2." class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ac08c701f1ad2b38d516d478bd5713b57c5e16" style="height: 3.343ex; vertical-align: -0.838ex; width: 37.99ex;" /></span></dd></dl><div class="toc" id="toc"><div class="toctitle" id="toctitle"><h2>Contents</h2></div></div><ul><li class="toclevel-1 tocsection-1"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Simple_example"><span class="tocnumber">1</span> <span class="toctext">Simple example</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Formal_introduction"><span class="tocnumber">2</span> <span class="toctext">Formal introduction</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Other_examples"><span class="tocnumber">3</span> <span class="toctext">Other examples</span></a><br />
<ul><li class="toclevel-2 tocsection-4"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Coordinate_transformation"><span class="tocnumber">3.1</span> <span class="toctext">Coordinate transformation</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Differentiation"><span class="tocnumber">3.2</span> <span class="toctext">Differentiation</span></a></li>
<li class="toclevel-2 tocsection-6"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Integration"><span class="tocnumber">3.3</span> <span class="toctext">Integration</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Differential_equations"><span class="tocnumber">3.4</span> <span class="toctext">Differential equations</span></a></li>
<li class="toclevel-2 tocsection-8"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Scaling_and_shifting"><span class="tocnumber">3.5</span> <span class="toctext">Scaling and shifting</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Momentum_vs._velocity"><span class="tocnumber">3.6</span> <span class="toctext">Momentum vs. velocity</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="https://en.wikipedia.org/wiki/Change_of_variables#Lagrangian_mechanics"><span class="tocnumber">3.7</span> <span class="toctext">Lagrangian mechanics</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-11"><a href="https://en.wikipedia.org/wiki/Change_of_variables#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a><span class="toctext"> </span></li>
</ul><br /><br />
#αντικατάσταση , #αλλαγήμεταβλητής , #ορισμός sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-89454644125981345322017-06-03T06:14:00.001-07:002017-06-03T06:14:27.815-07:00Αντικατάσταση: Πολυωνυμικές εξισώσεις, Ολοκληρώματα, The Why and How of Substitution : nrich.maths.org<a href="https://nrich.maths.org/7075">The Why and How of Substitution : nrich.maths.org</a><br /><br />
<br /><br />
<div id="staticbanner"><h1 class="headh4">The Why and How of Substitution</h1></div><h5 class="contdifflevel"><span class="contdifftext">Stage: 5</span> </h5><span class="author">Article by Alan and Toni Beardon</span><div id="datepub">Published October 2010,February 2011.</div><br /><br />
<span style="font-weight: bold;">Introduction</span><br /><br />
<br /><br />
"How could I have seen that?" This is a common response to seeing a <br />
substitution in mathematics, and this article attempts to answer this <br />
question. Sadly, the technique of substitution is often presented <br />
without mentioning the general idea behind <span style="font-style: italic;">all</span> substitutions. The effective use of substitution depends on two things: first, given a situation in which<br />
variables occur, a substitution is nothing more than <span style="font-style: italic;">a change of variable</span>; second, it is only effective if the change of variable <span style="font-style: italic;">simplifies the situation</span> and, hopefully, enables one to solve the simplified problem.<br /><br />
<br /><br />
There is no easy route to this: substitution will only work if the the <br />
original situation has some kind of symmetry or special property that we<br />
can exploit, and the skill in using the method of substitution depends <br />
on noticing this. Thus we should always be <span style="font-style: italic;">looking for special features</span> in the problem, and then <span style="font-style: italic;">be prepared<br />
to change the variable(s)</span> to exploit these features. Of course, <br />
once we have solved the problem in the new variables we have to rewrite <br />
the solution in terms of the original variables.<br /><br />
<br /><br />
The main idea behind substitution, then, is this. We are given some expression, or equation or graph involving the variable <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-1-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>. We make the substitution <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-2-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-4"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 71px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-5"><span class="mi" id="MathJax-Span-6" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-7" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-8" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">f<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-9" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-10" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-11" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>, and we now have a new expression, equation or graph involving the given terms, the variable <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-3-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-12"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 7px;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-13"><span class="mi" id="MathJax-Span-14" style="font-family: MathJax_Math; font-style: italic;">t</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.005em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> and the function <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-4-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-15"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 11px;"><span style="clip: rect(1.44em, 1000em, 2.656em, -0.405em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-16"><span class="mi" id="MathJax-Span-17" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.06em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.376em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span>. Since we are free to choose <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-5-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-18"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 11px;"><span style="clip: rect(1.44em, 1000em, 2.656em, -0.405em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-19"><span class="mi" id="MathJax-Span-20" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.06em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.376em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span> to be any function we like, it is highly likely that for a suitable choice of<br />
<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-6-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-21"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 11px;"><span style="clip: rect(1.44em, 1000em, 2.656em, -0.405em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-22"><span class="mi" id="MathJax-Span-23" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.06em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.376em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span> the new expression in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-7-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-24"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 7px;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-25"><span class="mi" id="MathJax-Span-26" style="font-family: MathJax_Math; font-style: italic;">t</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.005em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> will be simpler than the original expression in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-8-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-27"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-28"><span class="mi" id="MathJax-Span-29" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>. The skill lies in the selection of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-9-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-30"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 11px;"><span style="clip: rect(1.44em, 1000em, 2.656em, -0.405em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-31"><span class="mi" id="MathJax-Span-32" style="font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.06em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.376em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span>; the rest is just the algebraic manipulation of the variables.<br /><br />
<br /><br />
Let us now look at some examples with these ideas in mind.<br /><br />
<br /><br />
<span style="font-weight: bold;">Example <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-10-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-33"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-34"><span class="mn" id="MathJax-Span-35" style="font-family: MathJax_Main;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.045em; overflow: hidden; vertical-align: -0.069em; width: 0px;"></span></span></nobr></span> Polynomial Equations</span><br /><br />
<br /><br />
Let us consider the <span style="font-weight: bold;">polynomial equation</span><br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-11-Frame"><nobr><span class="math" id="MathJax-Span-36"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 286px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-37"><span class="mo" id="MathJax-Span-38" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-39" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-40" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-41" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span><span class="mo" id="MathJax-Span-42" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-43" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-44" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-45" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-46" style="font-family: MathJax_Main; padding-left: 0.222em;">4</span><span class="mo" id="MathJax-Span-47" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-48" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-49" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-50" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-51" style="font-family: MathJax_Main; padding-left: 0.222em;">6</span><span class="mo" id="MathJax-Span-52" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-53" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-54" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-55" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-56" style="font-family: MathJax_Main; padding-left: 0.222em;">9</span><span class="mo" id="MathJax-Span-57" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-58" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-59" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">a</span><span class="mo" id="MathJax-Span-60" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div><br /><br />
If we expand the left hand side we get a quartic in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-12-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-61"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-62"><span class="mi" id="MathJax-Span-63" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> which we cannot solve. However, we notice that the left hand side has a certain symmetry, namely <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-13-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-64"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 114px;"><span style="clip: rect(1.468em, 1000em, 2.533em, -0.377em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-65"><span class="mn" id="MathJax-Span-66" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-67" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-68" style="font-family: MathJax_Main; padding-left: 0.222em;">9</span><span class="mo" id="MathJax-Span-69" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-70" style="font-family: MathJax_Main; padding-left: 0.278em;">4</span><span class="mo" id="MathJax-Span-71" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-72" style="font-family: MathJax_Main; padding-left: 0.222em;">6</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.171em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span>. The roots of the left hand side are symmetric about the value <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-14-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-73"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.41em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-74"><span class="mn" id="MathJax-Span-75" style="font-family: MathJax_Main;">5</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span>, and this suggests that we should make a substitution that exploits this fact. Let us try <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-15-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-76"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 81px;"><span style="clip: rect(1.479em, 1000em, 2.533em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-77"><span class="mi" id="MathJax-Span-78" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-79" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-80" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">s</span><span class="mo" id="MathJax-Span-81" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-82" style="font-family: MathJax_Main; padding-left: 0.222em;">5</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.156em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span>; that is, we change the variable so that the symmetry is now about the<br />
origin (after all, <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-16-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-83"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 44px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.41em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-84"><span class="mn" id="MathJax-Span-85" style="font-family: MathJax_Main;">5</span><span class="mo" id="MathJax-Span-86" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-87" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-17-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-88"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 44px;"><span style="clip: rect(1.479em, 1000em, 2.533em, -0.41em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-89"><span class="mn" id="MathJax-Span-90" style="font-family: MathJax_Main;">5</span><span class="mo" id="MathJax-Span-91" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-92" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.156em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span> looks better than <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-18-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-93"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.468em, 1000em, 2.451em, -0.432em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-94"><span class="mn" id="MathJax-Span-95" style="font-family: MathJax_Main;">4</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.06em; overflow: hidden; vertical-align: -0.069em; width: 0px;"></span></span></nobr></span> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-19-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-96"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.418em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-97"><span class="mn" id="MathJax-Span-98" style="font-family: MathJax_Main;">6</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span>). With this we have<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-20-Frame"><nobr><span class="math" id="MathJax-Span-99"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 276px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-100"><span class="mo" id="MathJax-Span-101" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-102" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-103" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-104" style="font-family: MathJax_Main; padding-left: 0.222em;">4</span><span class="mo" id="MathJax-Span-105" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-106" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-107" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-108" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-109" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span><span class="mo" id="MathJax-Span-110" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-111" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-112" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-113" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-114" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span><span class="mo" id="MathJax-Span-115" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-116" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-117" style="font-family: MathJax_Math; font-style: italic;">s</span><span class="mo" id="MathJax-Span-118" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-119" style="font-family: MathJax_Main; padding-left: 0.222em;">4</span><span class="mo" id="MathJax-Span-120" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-121" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-122" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">a</span><span class="mo" id="MathJax-Span-123" style="font-family: MathJax_Main;">,</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div><br /><br />
or <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-21-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-124"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 181px;"><span style="clip: rect(1.311em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-125"><span class="mo" id="MathJax-Span-126" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-127"><span style="display: inline-block; height: 0px; position: relative; width: 0.892em;"><span style="clip: rect(1.703em, 1000em, 2.461em, -0.407em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-128" style="font-family: MathJax_Math; font-style: italic;">s</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-129" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-130" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-131" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span><span class="mo" id="MathJax-Span-132" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-133" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-134"><span style="display: inline-block; height: 0px; position: relative; width: 0.892em;"><span style="clip: rect(1.703em, 1000em, 2.461em, -0.407em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-135" style="font-family: MathJax_Math; font-style: italic;">s</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-136" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-137" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-138" style="font-family: MathJax_Main; padding-left: 0.222em;">16</span><span class="mo" id="MathJax-Span-139" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-140" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-141" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">a</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.613em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>. This is a quadratic equation in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-22-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-142"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 17px;"><span style="clip: rect(1.26em, 1000em, 2.41em, -0.407em); left: 0em; position: absolute; top: -2.247em;"><span class="mrow" id="MathJax-Span-143"><span class="msubsup" id="MathJax-Span-144"><span style="display: inline-block; height: 0px; position: relative; width: 0.892em;"><span style="clip: rect(1.703em, 1000em, 2.461em, -0.407em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-145" style="font-family: MathJax_Math; font-style: italic;">s</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-146" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.286em; overflow: hidden; vertical-align: -0.083em; width: 0px;"></span></span></nobr></span> which we can solve to give two values of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-23-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-147"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 17px;"><span style="clip: rect(1.26em, 1000em, 2.41em, -0.407em); left: 0em; position: absolute; top: -2.247em;"><span class="mrow" id="MathJax-Span-148"><span class="msubsup" id="MathJax-Span-149"><span style="display: inline-block; height: 0px; position: relative; width: 0.892em;"><span style="clip: rect(1.703em, 1000em, 2.461em, -0.407em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-150" style="font-family: MathJax_Math; font-style: italic;">s</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-151" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.286em; overflow: hidden; vertical-align: -0.083em; width: 0px;"></span></span></nobr></span> and four values of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-24-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-152"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 9px;"><span style="clip: rect(1.703em, 1000em, 2.461em, -0.407em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-153"><span class="mi" id="MathJax-Span-154" style="font-family: MathJax_Math; font-style: italic;">s</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.754em; overflow: hidden; vertical-align: -0.083em; width: 0px;"></span></span></nobr></span><br />
corresponding to the four solutions to the original equation. However, <br />
we can also simplify it with another substitution. The numbers <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-25-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-155"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-156"><span class="mn" id="MathJax-Span-157" style="font-family: MathJax_Main;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.045em; overflow: hidden; vertical-align: -0.069em; width: 0px;"></span></span></nobr></span> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-26-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-158"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 20px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.377em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-159"><span class="mn" id="MathJax-Span-160" style="font-family: MathJax_Main;">16</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> are symmetric about <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-27-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-161"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 40px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.377em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-162"><span class="mn" id="MathJax-Span-163" style="font-family: MathJax_Main;">17</span><span class="texatom" id="MathJax-Span-164"><span class="mrow" id="MathJax-Span-165"><span class="mo" id="MathJax-Span-166" style="font-family: MathJax_Main;">/</span></span></span><span class="mn" id="MathJax-Span-167" style="font-family: MathJax_Main;">2</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span> so we now make the substitution <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-28-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-168"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 96px;"><span style="clip: rect(1.245em, 1000em, 2.828em, -0.407em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-169"><span class="msubsup" id="MathJax-Span-170"><span style="display: inline-block; height: 0px; position: relative; width: 0.892em;"><span style="clip: rect(1.703em, 1000em, 2.461em, -0.407em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-171" style="font-family: MathJax_Math; font-style: italic;">s</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-172" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-173" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-174" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">t</span><span class="mo" id="MathJax-Span-175" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-176" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.835em;"><span style="clip: rect(1.616em, 1000em, 2.415em, -0.401em); left: 50%; margin-left: -0.357em; position: absolute; top: -2.668em;"><span class="mn" id="MathJax-Span-177" style="font-family: MathJax_Main; font-size: 70.7%;">17</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(1.623em, 1000em, 2.4em, -0.424em); left: 50%; margin-left: -0.179em; position: absolute; top: -1.87em;"><span class="mn" id="MathJax-Span-178" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.835em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.874em; overflow: hidden; vertical-align: -0.582em; width: 0px;"></span></span></nobr></span>. This gives <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-29-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-179"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 105px;"><span style="clip: rect(1.252em, 1000em, 2.835em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-180"><span class="msubsup" id="MathJax-Span-181"><span style="display: inline-block; height: 0px; position: relative; width: 0.79em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-182" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-183" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-184" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-185" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">a</span><span class="mo" id="MathJax-Span-186" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-187" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.192em;"><span style="clip: rect(1.623em, 1000em, 2.415em, -0.424em); left: 50%; margin-left: -0.536em; position: absolute; top: -2.668em;"><span class="mn" id="MathJax-Span-188" style="font-family: MathJax_Main; font-size: 70.7%;">225</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(1.615em, 1000em, 2.4em, -0.44em); left: 50%; margin-left: -0.179em; position: absolute; top: -1.862em;"><span class="mn" id="MathJax-Span-189" style="font-family: MathJax_Main; font-size: 70.7%;">4</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.192em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.875em; overflow: hidden; vertical-align: -0.592em; width: 0px;"></span></span></nobr></span><br />
so that <span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-30-Frame"><nobr><span class="math" id="MathJax-Span-190"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 248px;"><span style="clip: rect(-2.656em, 1000em, 6.65em, -0.274em); left: 0em; position: absolute; top: -2.247em;"><span class="mrow" id="MathJax-Span-191"><span class="mtable" id="MathJax-Span-192" style="padding-left: 0.167em; padding-right: 0.167em;"><span style="display: inline-block; height: 0px; position: relative; width: 12.351em;"><span style="clip: rect(3.89em, 1000em, 11.109em, -0.441em); left: 0em; position: absolute; top: -7.915em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.613em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); position: absolute; right: 0em; top: -5.543em;"><span class="mtd" id="MathJax-Span-193"><span class="mrow" id="MathJax-Span-194"><span class="mi" id="MathJax-Span-195" style="font-family: MathJax_Math; font-style: italic;">t</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.703em, 1000em, 2.461em, -0.407em); position: absolute; right: 0em; top: -2.567em;"><span class="mtd" id="MathJax-Span-210"><span class="mrow" id="MathJax-Span-211"><span class="mi" id="MathJax-Span-212" style="font-family: MathJax_Math; font-style: italic;">s</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); position: absolute; right: 0em; top: 0.733em;"><span class="mtd" id="MathJax-Span-233"><span class="mrow" id="MathJax-Span-234"><span class="mi" id="MathJax-Span-235" style="font-family: MathJax_Math; font-style: italic;">x</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="display: inline-block; height: 7.915em; width: 0px;"></span></span><span style="clip: rect(3.843em, 1000em, 10.759em, -0.404em); left: 0.891em; position: absolute; top: -7.608em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.766em;"><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: -5.543em;"><span class="mtd" id="MathJax-Span-196"><span class="mrow" id="MathJax-Span-197"><span class="mo" id="MathJax-Span-198" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: -2.567em;"><span class="mtd" id="MathJax-Span-213"><span class="mrow" id="MathJax-Span-214"><span class="mo" id="MathJax-Span-215" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: 0.733em;"><span class="mtd" id="MathJax-Span-236"><span class="mrow" id="MathJax-Span-237"><span class="mo" id="MathJax-Span-238" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="display: inline-block; height: 7.608em; width: 0px;"></span></span><span style="clip: rect(5.105em, 1000em, 14.411em, -0.41em); left: 1.934em; position: absolute; top: -10.008em;"><span style="display: inline-block; height: 0px; position: relative; width: 10.417em;"><span style="clip: rect(0.64em, 1000em, 3.347em, -0.404em); left: 0em; position: absolute; top: -5.543em;"><span class="mtd" id="MathJax-Span-199"><span class="mrow" id="MathJax-Span-200"><span class="mo" id="MathJax-Span-201" style="font-family: MathJax_Main;">±</span><span class="msqrt" id="MathJax-Span-202"><span style="display: inline-block; height: 0px; position: relative; width: 4.723em;"><span style="clip: rect(0.802em, 1000em, 3.137em, -0.427em); left: 1.021em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-203"><span class="mi" id="MathJax-Span-204" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="mo" id="MathJax-Span-205" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-206" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.652em;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.41em); left: 50%; margin-left: -0.766em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-207" style="font-family: MathJax_Main;">225</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.468em, 1000em, 2.451em, -0.432em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-208" style="font-family: MathJax_Main;">4</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.652em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.56em, 1000em, 4.006em, -0.46em); left: 1.021em; position: absolute; top: -5.217em;"><span style="display: inline-block; height: 0px; position: relative; width: 3.701em;"><span style="font-family: MathJax_Main; left: -0.084em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 3.007em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.41em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.93em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.449em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.969em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 2.488em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(2.38em, 1000em, 5.086em, -0.349em); left: 0em; position: absolute; top: -4.037em;"><span style="font-family: MathJax_Size3;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-209" style="font-family: MathJax_Main;">,</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.364em, 1000em, 3.67em, -0.404em); left: 0em; position: absolute; top: -2.567em;"><span class="mtd" id="MathJax-Span-216"><span class="mrow" id="MathJax-Span-217"><span class="mo" id="MathJax-Span-218" style="font-family: MathJax_Main;">±</span><span class="msqrt" id="MathJax-Span-219"><span style="display: inline-block; height: 0px; position: relative; width: 8.399em;"><span style="clip: rect(0.64em, 1000em, 3.347em, -0.34em); left: 1.021em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-220"><span class="mfrac" id="MathJax-Span-221" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.141em;"><span style="clip: rect(1.469em, 1000em, 2.473em, -0.377em); left: 50%; margin-left: -0.511em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-222" style="font-family: MathJax_Main;">17</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-223" style="font-family: MathJax_Main;">2</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.141em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-224" style="font-family: MathJax_Main; padding-left: 0.222em;">±</span><span class="msqrt" id="MathJax-Span-225" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 4.723em;"><span style="clip: rect(0.802em, 1000em, 3.137em, -0.427em); left: 1.021em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-226"><span class="mi" id="MathJax-Span-227" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="mo" id="MathJax-Span-228" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-229" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.652em;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.41em); left: 50%; margin-left: -0.766em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-230" style="font-family: MathJax_Main;">225</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.468em, 1000em, 2.451em, -0.432em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-231" style="font-family: MathJax_Main;">4</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.652em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.56em, 1000em, 4.006em, -0.46em); left: 1.021em; position: absolute; top: -5.217em;"><span style="display: inline-block; height: 0px; position: relative; width: 3.701em;"><span style="font-family: MathJax_Main; left: -0.084em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 3.007em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.41em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.93em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.449em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.969em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 2.488em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(2.38em, 1000em, 5.086em, -0.349em); left: 0em; position: absolute; top: -4.037em;"><span style="font-family: MathJax_Size3;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.56em, 1000em, 4.006em, -0.46em); left: 1.021em; position: absolute; top: -5.493em;"><span style="display: inline-block; height: 0px; position: relative; width: 7.378em;"><span style="font-family: MathJax_Main; left: -0.084em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 6.684em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.414em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.936em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.459em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.981em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 2.504em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 3.026em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 3.549em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 4.071em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 4.594em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 5.116em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 5.639em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 6.161em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(2.08em, 1000em, 5.386em, -0.349em); left: 0em; position: absolute; top: -4.013em;"><span style="font-family: MathJax_Size4;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-232" style="font-family: MathJax_Main;">,</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.364em, 1000em, 3.67em, -0.41em); left: 0em; position: absolute; top: 0.733em;"><span class="mtd" id="MathJax-Span-239"><span class="mrow" id="MathJax-Span-240"><span class="mn" id="MathJax-Span-241" style="font-family: MathJax_Main;">5</span><span class="mo" id="MathJax-Span-242" style="font-family: MathJax_Main; padding-left: 0.222em;">±</span><span class="msqrt" id="MathJax-Span-243" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 8.399em;"><span style="clip: rect(0.64em, 1000em, 3.347em, -0.34em); left: 1.021em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-244"><span class="mfrac" id="MathJax-Span-245" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.141em;"><span style="clip: rect(1.469em, 1000em, 2.473em, -0.377em); left: 50%; margin-left: -0.511em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-246" style="font-family: MathJax_Main;">17</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-247" style="font-family: MathJax_Main;">2</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.141em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-248" style="font-family: MathJax_Main; padding-left: 0.222em;">±</span><span class="msqrt" id="MathJax-Span-249" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 4.723em;"><span style="clip: rect(0.802em, 1000em, 3.137em, -0.427em); left: 1.021em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-250"><span class="mi" id="MathJax-Span-251" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="mo" id="MathJax-Span-252" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-253" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.652em;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.41em); left: 50%; margin-left: -0.766em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-254" style="font-family: MathJax_Main;">225</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.468em, 1000em, 2.451em, -0.432em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-255" style="font-family: MathJax_Main;">4</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.652em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.56em, 1000em, 4.006em, -0.46em); left: 1.021em; position: absolute; top: -5.217em;"><span style="display: inline-block; height: 0px; position: relative; width: 3.701em;"><span style="font-family: MathJax_Main; left: -0.084em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 3.007em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.41em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.93em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.449em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.969em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 2.488em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(2.38em, 1000em, 5.086em, -0.349em); left: 0em; position: absolute; top: -4.037em;"><span style="font-family: MathJax_Size3;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.56em, 1000em, 4.006em, -0.46em); left: 1.021em; position: absolute; top: -5.493em;"><span style="display: inline-block; height: 0px; position: relative; width: 7.378em;"><span style="font-family: MathJax_Main; left: -0.084em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 6.684em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.414em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.936em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.459em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 1.981em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 2.504em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 3.026em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 3.549em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 4.071em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 4.594em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 5.116em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 5.639em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 6.161em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(2.08em, 1000em, 5.386em, -0.349em); left: 0em; position: absolute; top: -4.013em;"><span style="font-family: MathJax_Size4;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-256" style="font-family: MathJax_Main;">.</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="display: inline-block; height: 10.008em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 12.379em; overflow: hidden; vertical-align: -5.849em; width: 0px;"></span></span></nobr></span></div><br /><br />
You should check that if we put <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-31-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-257"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 47px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-258"><span class="mi" id="MathJax-Span-259" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="mo" id="MathJax-Span-260" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-261" style="font-family: MathJax_Main; padding-left: 0.278em;">0</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> in this formula we do get the expected solutions <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-32-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-262"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 66px;"><span style="clip: rect(1.468em, 1000em, 2.645em, -0.377em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-263"><span class="mn" id="MathJax-Span-264" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-265" style="font-family: MathJax_Main;">,</span><span class="mn" id="MathJax-Span-266" style="font-family: MathJax_Main; padding-left: 0.167em;">4</span><span class="mo" id="MathJax-Span-267" style="font-family: MathJax_Main;">,</span><span class="mn" id="MathJax-Span-268" style="font-family: MathJax_Main; padding-left: 0.167em;">6</span><span class="mo" id="MathJax-Span-269" style="font-family: MathJax_Main;">,</span><span class="mn" id="MathJax-Span-270" style="font-family: MathJax_Main; padding-left: 0.167em;">9</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.323em; overflow: hidden; vertical-align: -0.333em; width: 0px;"></span></span></nobr></span>. What do you get if <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-33-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-271"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 67px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-272"><span class="mi" id="MathJax-Span-273" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="mo" id="MathJax-Span-274" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-275" style="font-family: MathJax_Main; padding-left: 0.278em;">216</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span>?<br /><br />
<br /><br />
<span style="font-weight: bold;">Example <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-34-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-276"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-277"><span class="mn" id="MathJax-Span-278" style="font-family: MathJax_Main;">2</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.045em; overflow: hidden; vertical-align: -0.069em; width: 0px;"></span></span></nobr></span> Rational functions</span><br /><br />
<br /><br />
We want to solve the following equation:<br /><br />
<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-35-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-279"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 184px;"><span style="clip: rect(1.091em, 1000em, 2.952em, -0.34em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-280"><span class="mfrac" id="MathJax-Span-281" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 3.745em;"><span style="clip: rect(1.504em, 1000em, 2.458em, -0.435em); left: 50%; margin-left: -1.813em; position: absolute; top: -2.71em;"><span class="mrow" id="MathJax-Span-282"><span class="msubsup" id="MathJax-Span-283"><span style="display: inline-block; height: 0px; position: relative; width: 0.717em;"><span style="clip: rect(1.781em, 1000em, 2.408em, -0.435em); left: 0em; position: absolute; top: -2.247em;"><span class="mi" id="MathJax-Span-284" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.503em;"><span class="mn" id="MathJax-Span-285" style="font-family: MathJax_Main; font-size: 50%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-286" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-287" style="font-family: MathJax_Main; font-size: 70.7%;">10</span><span class="mi" id="MathJax-Span-288" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span><span class="mo" id="MathJax-Span-289" style="font-family: MathJax_Main; font-size: 70.7%;">+</span><span class="mn" id="MathJax-Span-290" style="font-family: MathJax_Main; font-size: 70.7%;">15</span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(1.556em, 1000em, 2.458em, -0.435em); left: 50%; margin-left: -1.634em; position: absolute; top: -1.804em;"><span class="mrow" id="MathJax-Span-291"><span class="msubsup" id="MathJax-Span-292"><span style="display: inline-block; height: 0px; position: relative; width: 0.717em;"><span style="clip: rect(1.781em, 1000em, 2.408em, -0.435em); left: 0em; position: absolute; top: -2.247em;"><span class="mi" id="MathJax-Span-293" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.451em;"><span class="mn" id="MathJax-Span-294" style="font-family: MathJax_Main; font-size: 50%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-295" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-296" style="font-family: MathJax_Main; font-size: 70.7%;">6</span><span class="mi" id="MathJax-Span-297" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span><span class="mo" id="MathJax-Span-298" style="font-family: MathJax_Main; font-size: 70.7%;">+</span><span class="mn" id="MathJax-Span-299" style="font-family: MathJax_Main; font-size: 70.7%;">15</span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 3.745em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-300" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mfrac" id="MathJax-Span-301" style="padding-left: 0.398em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 3.388em;"><span style="clip: rect(1.623em, 1000em, 2.415em, -0.43em); left: 50%; margin-left: -0.383em; position: absolute; top: -2.668em;"><span class="mrow" id="MathJax-Span-302"><span class="mn" id="MathJax-Span-303" style="font-family: MathJax_Main; font-size: 70.7%;">3</span><span class="mi" id="MathJax-Span-304" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(1.556em, 1000em, 2.458em, -0.435em); left: 50%; margin-left: -1.634em; position: absolute; top: -1.804em;"><span class="mrow" id="MathJax-Span-305"><span class="msubsup" id="MathJax-Span-306"><span style="display: inline-block; height: 0px; position: relative; width: 0.717em;"><span style="clip: rect(1.781em, 1000em, 2.408em, -0.435em); left: 0em; position: absolute; top: -2.247em;"><span class="mi" id="MathJax-Span-307" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.451em;"><span class="mn" id="MathJax-Span-308" style="font-family: MathJax_Main; font-size: 50%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-309" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-310" style="font-family: MathJax_Main; font-size: 70.7%;">8</span><span class="mi" id="MathJax-Span-311" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span><span class="mo" id="MathJax-Span-312" style="font-family: MathJax_Main; font-size: 70.7%;">+</span><span class="mn" id="MathJax-Span-313" style="font-family: MathJax_Main; font-size: 70.7%;">15</span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 3.388em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-314" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 2.253em; overflow: hidden; vertical-align: -0.751em; width: 0px;"></span></span></nobr></span><br /><br />
By clearing fractions this becomes a quartic equation which is difficult to solve. Observing the occurrences of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-36-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-315"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 64px;"><span style="clip: rect(1.311em, 1000em, 2.533em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-316"><span class="msubsup" id="MathJax-Span-317"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-318" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-319" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-320" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-321" style="font-family: MathJax_Main; padding-left: 0.222em;">15</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.384em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span>, and the symmetry of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-37-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-322"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 110px;"><span style="clip: rect(1.311em, 1000em, 2.533em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-323"><span class="msubsup" id="MathJax-Span-324"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-325" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-326" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-327" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-328" style="font-family: MathJax_Main; padding-left: 0.222em;">6</span><span class="mi" id="MathJax-Span-329" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-330" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-331" style="font-family: MathJax_Main; padding-left: 0.222em;">15</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.384em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span>, <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-38-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-332"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 110px;"><span style="clip: rect(1.311em, 1000em, 2.533em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-333"><span class="msubsup" id="MathJax-Span-334"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-335" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-336" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-337" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-338" style="font-family: MathJax_Main; padding-left: 0.222em;">8</span><span class="mi" id="MathJax-Span-339" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-340" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-341" style="font-family: MathJax_Main; padding-left: 0.222em;">15</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.384em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-39-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-342"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 120px;"><span style="clip: rect(1.311em, 1000em, 2.533em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-343"><span class="msubsup" id="MathJax-Span-344"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-345" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-346" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-347" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-348" style="font-family: MathJax_Main; padding-left: 0.222em;">10</span><span class="mi" id="MathJax-Span-349" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-350" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-351" style="font-family: MathJax_Main; padding-left: 0.222em;">15</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.384em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span>, we can turn this into a quadratic equation by substituting <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-40-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-352"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 124px;"><span style="clip: rect(1.252em, 1000em, 2.804em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-353"><span class="mi" id="MathJax-Span-354" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-355" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-356" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">x</span><span class="mo" id="MathJax-Span-357" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-358" style="font-family: MathJax_Main; padding-left: 0.222em;">8</span><span class="mo" id="MathJax-Span-359" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-360" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.835em;"><span style="clip: rect(1.623em, 1000em, 2.415em, -0.401em); left: 50%; margin-left: -0.357em; position: absolute; top: -2.668em;"><span class="mn" id="MathJax-Span-361" style="font-family: MathJax_Main; font-size: 70.7%;">15</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(1.781em, 1000em, 2.408em, -0.435em); left: 50%; margin-left: -0.204em; position: absolute; top: -1.902em;"><span class="mi" id="MathJax-Span-362" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.835em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.832em; overflow: hidden; vertical-align: -0.549em; width: 0px;"></span></span></nobr></span>. We get the equation<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-41-Frame"><nobr><span class="math" id="MathJax-Span-363"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 100px;"><span style="clip: rect(0.802em, 1000em, 3.219em, -0.34em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-364"><span class="mfrac" id="MathJax-Span-365" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 2.214em;"><span style="clip: rect(1.479em, 1000em, 2.462em, -0.441em); left: 50%; margin-left: -1.047em; position: absolute; top: -2.974em;"><span class="mrow" id="MathJax-Span-366"><span class="mi" id="MathJax-Span-367" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-368" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-369" style="font-family: MathJax_Main; padding-left: 0.222em;">2</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.533em, -0.441em); left: 50%; margin-left: -1.047em; position: absolute; top: -1.612em;"><span class="mrow" id="MathJax-Span-370"><span class="mi" id="MathJax-Span-371" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-372" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-373" style="font-family: MathJax_Main; padding-left: 0.222em;">2</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 2.214em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-374" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mfrac" id="MathJax-Span-375" style="padding-left: 0.398em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.48em, 1000em, 2.473em, -0.418em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-376" style="font-family: MathJax_Main;">3</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 50%; margin-left: -0.179em; position: absolute; top: -1.612em;"><span class="mi" id="MathJax-Span-377" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-378" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 3.009em; overflow: hidden; vertical-align: -1.114em; width: 0px;"></span></span></nobr></span></div><br /><br />
This simplifies to <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-42-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-379"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 127px;"><span style="clip: rect(1.311em, 1000em, 2.473em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-380"><span class="msubsup" id="MathJax-Span-381"><span style="display: inline-block; height: 0px; position: relative; width: 0.79em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-382" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-383" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-384" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-385" style="font-family: MathJax_Main; padding-left: 0.222em;">5</span><span class="mi" id="MathJax-Span-386" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-387" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-388" style="font-family: MathJax_Main; padding-left: 0.222em;">6</span><span class="mo" id="MathJax-Span-389" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-390" style="font-family: MathJax_Main; padding-left: 0.278em;">0</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.303em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span>, so that <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-43-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-391"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 7px;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-392"><span class="mi" id="MathJax-Span-393" style="font-family: MathJax_Math; font-style: italic;">t</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.005em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> is <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-44-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-394"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.418em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-395"><span class="mn" id="MathJax-Span-396" style="font-family: MathJax_Main;">6</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> or <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-45-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-397"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 25px;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.376em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-398"><span class="mo" id="MathJax-Span-399" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-400" style="font-family: MathJax_Main;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.045em; overflow: hidden; vertical-align: -0.069em; width: 0px;"></span></span></nobr></span>. Each value of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-46-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-401"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 7px;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-402"><span class="mi" id="MathJax-Span-403" style="font-family: MathJax_Math; font-style: italic;">t</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.005em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> gives a quadratic equation in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-47-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-404"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-405"><span class="mi" id="MathJax-Span-406" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>, giving four solutions of the original equation in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-48-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-407"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-408"><span class="mi" id="MathJax-Span-409" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>. The two quadratic equations are <span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-49-Frame"><nobr><span class="math" id="MathJax-Span-410"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 334px;"><span style="clip: rect(0.802em, 1000em, 3.148em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-411"><span class="mi" id="MathJax-Span-412" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-413" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-414" style="font-family: MathJax_Main; padding-left: 0.222em;">8</span><span class="mo" id="MathJax-Span-415" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-416" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.141em;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.377em); left: 50%; margin-left: -0.511em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-417" style="font-family: MathJax_Main;">15</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 50%; margin-left: -0.306em; position: absolute; top: -1.612em;"><span class="mi" id="MathJax-Span-418" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.141em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-419" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-420" style="font-family: MathJax_Main; padding-left: 0.278em;">6</span><span class="mo" id="MathJax-Span-421" style="font-family: MathJax_Main;">,</span><span class="mspace" id="MathJax-Span-422" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 2em;"></span><span class="mi" id="MathJax-Span-423" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">x</span><span class="mo" id="MathJax-Span-424" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-425" style="font-family: MathJax_Main; padding-left: 0.222em;">8</span><span class="mo" id="MathJax-Span-426" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-427" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.141em;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.377em); left: 50%; margin-left: -0.511em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-428" style="font-family: MathJax_Main;">15</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 50%; margin-left: -0.306em; position: absolute; top: -1.612em;"><span class="mi" id="MathJax-Span-429" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.141em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-430" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mo" id="MathJax-Span-431" style="font-family: MathJax_Main; padding-left: 0.278em;">−</span><span class="mn" id="MathJax-Span-432" style="font-family: MathJax_Main;">1.</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 2.913em; overflow: hidden; vertical-align: -1.017em; width: 0px;"></span></span></nobr></span></div><br /><br />
These equations simplify to<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-50-Frame"><nobr><span class="math" id="MathJax-Span-433"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 353px;"><span style="clip: rect(1.261em, 1000em, 2.645em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-434"><span class="msubsup" id="MathJax-Span-435"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-436" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-437" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-438" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-439" style="font-family: MathJax_Main; padding-left: 0.222em;">14</span><span class="mi" id="MathJax-Span-440" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-441" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-442" style="font-family: MathJax_Main; padding-left: 0.222em;">15</span><span class="mo" id="MathJax-Span-443" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-444" style="font-family: MathJax_Main; padding-left: 0.222em;">0</span><span class="mo" id="MathJax-Span-445" style="font-family: MathJax_Main;">,</span><span class="mspace" id="MathJax-Span-446" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 2em;"></span><span class="msubsup" id="MathJax-Span-447" style="padding-left: 0.167em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-448" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-449" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-450" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-451" style="font-family: MathJax_Main; padding-left: 0.222em;">7</span><span class="mi" id="MathJax-Span-452" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-453" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-454" style="font-family: MathJax_Main; padding-left: 0.222em;">15</span><span class="mo" id="MathJax-Span-455" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-456" style="font-family: MathJax_Main; padding-left: 0.278em;">0</span><span class="mo" id="MathJax-Span-457" style="font-family: MathJax_Main;">,</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.605em; overflow: hidden; vertical-align: -0.333em; width: 0px;"></span></span></nobr></span></div><br /><br />
and the four solutions are <span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-51-Frame"><nobr><span class="math" id="MathJax-Span-458"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 223px;"><span style="clip: rect(0.904em, 1000em, 3.239em, -0.405em); left: 0em; position: absolute; top: -2.4em;"><span class="mrow" id="MathJax-Span-459"><span class="mn" id="MathJax-Span-460" style="font-family: MathJax_Main;">7</span><span class="mo" id="MathJax-Span-461" style="font-family: MathJax_Main; padding-left: 0.222em;">±</span><span class="msqrt" id="MathJax-Span-462" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.947em;"><span style="clip: rect(1.468em, 1000em, 2.473em, -0.418em); left: 0.817em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-463"><span class="mn" id="MathJax-Span-464" style="font-family: MathJax_Main;">34</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.56em, 1000em, 4.006em, -0.46em); left: 0.817em; position: absolute; top: -4.517em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.13em;"><span style="font-family: MathJax_Main; left: -0.084em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.436em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(3.03em, 1000em, 4.336em, -0.388em); left: 0em; position: absolute; top: -3.987em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-465" style="font-family: MathJax_Main;">,</span><span class="mspace" id="MathJax-Span-466" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 1em;"></span><span class="mfrac" id="MathJax-Span-467" style="padding-left: 0.287em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-468" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-469" style="font-family: MathJax_Main;">2</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-470" style="padding-left: 0.167em;"><span class="mrow" id="MathJax-Span-471"><span class="mo" id="MathJax-Span-472" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="mn" id="MathJax-Span-473" style="font-family: MathJax_Main;">7</span><span class="mo" id="MathJax-Span-474" style="font-family: MathJax_Main; padding-left: 0.222em;">±</span><span class="mi" id="MathJax-Span-475" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">i</span><span class="msqrt" id="MathJax-Span-476"><span style="display: inline-block; height: 0px; position: relative; width: 1.947em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 0.817em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-477"><span class="mn" id="MathJax-Span-478" style="font-family: MathJax_Main;">11</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.56em, 1000em, 4.006em, -0.46em); left: 0.817em; position: absolute; top: -4.522em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.13em;"><span style="font-family: MathJax_Main; left: -0.084em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.436em; position: absolute; top: -3.983em;">−<span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(3.03em, 1000em, 4.336em, -0.388em); left: 0em; position: absolute; top: -3.992em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-479"><span class="mrow" id="MathJax-Span-480"><span class="mo" id="MathJax-Span-481" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">)</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="mo" id="MathJax-Span-482" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; height: 2.4em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 2.898em; overflow: hidden; vertical-align: -1.002em; width: 0px;"></span></span></nobr></span></div><br /><br />
<br /><br />
<span style="font-weight: bold;">Example <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-52-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-483"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.48em, 1000em, 2.473em, -0.418em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-484"><span class="mn" id="MathJax-Span-485" style="font-family: MathJax_Main;">3</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.073em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> Integration by substitution</span> <br /><br />
<br /><br />
Evaluate <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-53-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-486"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 181px;"><span style="clip: rect(1.239em, 1000em, 2.859em, -0.434em); left: 0em; position: absolute; top: -2.4em;"><span class="mrow" id="MathJax-Span-487"><span class="mi" id="MathJax-Span-488" style="font-family: MathJax_Math; font-style: italic;">I<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.064em;"></span></span><span class="mo" id="MathJax-Span-489" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mo" id="MathJax-Span-490" style="font-family: MathJax_Size1; padding-left: 0.278em; vertical-align: 0em;">∫<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.138em;"></span></span><span class="mo" id="MathJax-Span-491" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-492" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-493" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-494" style="font-family: MathJax_Main; padding-left: 0.222em;">9</span><span class="msubsup" id="MathJax-Span-495"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-496" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-497" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-498"><span style="display: inline-block; height: 0px; position: relative; width: 1.556em;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.405em); left: 0em; position: absolute; top: -2.298em;"><span class="mo" id="MathJax-Span-499" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.724em;"><span class="texatom" id="MathJax-Span-500"><span class="mrow" id="MathJax-Span-501"><span class="mn" id="MathJax-Span-502" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span class="texatom" id="MathJax-Span-503"><span class="mrow" id="MathJax-Span-504"><span class="mo" id="MathJax-Span-505" style="font-family: MathJax_Main; font-size: 70.7%;">/</span></span></span><span class="mn" id="MathJax-Span-506" style="font-family: MathJax_Main; font-size: 70.7%;">2</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mspace" id="MathJax-Span-507" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 0.167em;"></span><span class="mi" id="MathJax-Span-508" style="font-family: MathJax_Math; font-style: italic;">d<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-509" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-510" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; height: 2.4em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.924em; overflow: hidden; vertical-align: -0.485em; width: 0px;"></span></span></nobr></span><br /><br />
Here a trigonometric substitution leads to a simpler integral. Because of the relation <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-54-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-511"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 157px;"><span style="clip: rect(1.278em, 1000em, 2.462em, -0.377em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-512"><span class="mn" id="MathJax-Span-513" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-514" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="msubsup" id="MathJax-Span-515" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.658em;"><span style="clip: rect(1.476em, 1000em, 2.462em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-516" style="font-family: MathJax_Main;">sin</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 1.225em; position: absolute; top: -2.643em;"><span class="mn" id="MathJax-Span-517" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-518"></span><span class="mi" id="MathJax-Span-519" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span><span class="mo" id="MathJax-Span-520" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="msubsup" id="MathJax-Span-521" style="padding-left: 0.278em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.76em;"><span style="clip: rect(1.697em, 1000em, 2.462em, -0.426em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-522" style="font-family: MathJax_Main;">cos</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 1.328em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-523" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-524"></span><span class="mi" id="MathJax-Span-525" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.333em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>, we substitute <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-55-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-526"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 86px;"><span style="clip: rect(1.476em, 1000em, 2.473em, -0.418em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-527"><span class="mn" id="MathJax-Span-528" style="font-family: MathJax_Main;">3</span><span class="mi" id="MathJax-Span-529" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-530" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-531" style="font-family: MathJax_Main; padding-left: 0.278em;">sin</span><span class="mo" id="MathJax-Span-532"></span><span class="mi" id="MathJax-Span-533" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.079em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-56-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-534"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 123px;"><span style="clip: rect(1.451em, 1000em, 2.473em, -0.418em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-535"><span class="mn" id="MathJax-Span-536" style="font-family: MathJax_Main;">3</span><span class="mi" id="MathJax-Span-537" style="font-family: MathJax_Math; font-style: italic;">d<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-538" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-539" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-540" style="font-family: MathJax_Main; padding-left: 0.278em;">cos</span><span class="mo" id="MathJax-Span-541"></span><span class="mi" id="MathJax-Span-542" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span><span class="mspace" id="MathJax-Span-543" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 0.167em;"></span><span class="mi" id="MathJax-Span-544" style="font-family: MathJax_Math; font-style: italic;">d<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-545" style="font-family: MathJax_Math; font-style: italic;">u</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.113em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> and get<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-57-Frame"><nobr><span class="math" id="MathJax-Span-546"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 317px;"><span style="clip: rect(-2.965em, 1000em, 6.958em, -0.293em); left: 0em; position: absolute; top: -2.247em;"><span class="mrow" id="MathJax-Span-547"><span class="mtable" id="MathJax-Span-548" style="padding-left: 0.167em; padding-right: 0.167em;"><span style="display: inline-block; height: 0px; position: relative; width: 15.874em;"><span style="clip: rect(4.709em, 1000em, 13.246em, -0.46em); left: 0em; position: absolute; top: -9.242em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.511em;"><span style="clip: rect(1.462em, 1000em, 2.451em, -0.434em); position: absolute; right: 0em; top: -5.995em;"><span class="mtd" id="MathJax-Span-549"><span class="mrow" id="MathJax-Span-550"><span class="mi" id="MathJax-Span-551" style="font-family: MathJax_Math; font-style: italic;">I<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.064em;"></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.83em, 1000em, 4.136em, -0.46em); position: absolute; right: 0em; top: -5.158em;"><span class="mtd" id="MathJax-Span-595"><span class="mrow" id="MathJax-Span-596"></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(3.83em, 1000em, 4.136em, -0.46em); position: absolute; right: 0em; top: -2.636em;"><span class="mtd" id="MathJax-Span-614"><span class="mrow" id="MathJax-Span-615"></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(3.83em, 1000em, 4.136em, -0.46em); position: absolute; right: 0em; top: -0.133em;"><span class="mtd" id="MathJax-Span-639"><span class="mrow" id="MathJax-Span-640"></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 9.242em; width: 0px;"></span></span><span style="clip: rect(4.667em, 1000em, 12.855em, -0.404em); left: 0.788em; position: absolute; top: -8.885em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.766em;"><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: -5.995em;"><span class="mtd" id="MathJax-Span-552"><span class="mrow" id="MathJax-Span-553"><span class="mo" id="MathJax-Span-554" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: -3.473em;"><span class="mtd" id="MathJax-Span-597"><span class="mrow" id="MathJax-Span-598"><span class="mo" id="MathJax-Span-599" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: -0.951em;"><span class="mtd" id="MathJax-Span-616"><span class="mrow" id="MathJax-Span-617"><span class="mo" id="MathJax-Span-618" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: 1.552em;"><span class="mtd" id="MathJax-Span-641"><span class="mrow" id="MathJax-Span-642"><span class="mo" id="MathJax-Span-643" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="display: inline-block; height: 8.885em; width: 0px;"></span></span><span style="clip: rect(5.409em, 1000em, 15.332em, -0.405em); left: 1.832em; position: absolute; top: -10.621em;"><span style="display: inline-block; height: 0px; position: relative; width: 14.042em;"><span style="clip: rect(1.754em, 1000em, 4.282em, -0.405em); left: 0em; position: absolute; top: -6.965em;"><span class="mtd" id="MathJax-Span-555"><span class="mrow" id="MathJax-Span-556"><span class="mo" id="MathJax-Span-557" style="font-family: MathJax_Size2; vertical-align: 0.001em;">∫<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.388em;"></span></span><span class="texatom" id="MathJax-Span-558" style="padding-left: 0.167em;"><span class="mrow" id="MathJax-Span-559"><span class="mo" id="MathJax-Span-560" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="mn" id="MathJax-Span-561" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-562" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="msubsup" id="MathJax-Span-563" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.658em;"><span style="clip: rect(1.476em, 1000em, 2.462em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-564" style="font-family: MathJax_Main;">sin</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 1.225em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-565" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-566"></span><span class="mi" id="MathJax-Span-567" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span><span class="msubsup" id="MathJax-Span-568"><span style="display: inline-block; height: 0px; position: relative; width: 1.607em;"><span style="clip: rect(1.397em, 1000em, 2.903em, -0.425em); left: 0em; position: absolute; top: -2.4em;"><span class="texatom" id="MathJax-Span-569"><span class="mrow" id="MathJax-Span-570"><span class="mo" id="MathJax-Span-571" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">)</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span style="display: inline-block; height: 2.4em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.823em;"><span class="texatom" id="MathJax-Span-572"><span class="mrow" id="MathJax-Span-573"><span class="mn" id="MathJax-Span-574" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span class="texatom" id="MathJax-Span-575"><span class="mrow" id="MathJax-Span-576"><span class="mo" id="MathJax-Span-577" style="font-family: MathJax_Main; font-size: 70.7%;">/</span></span></span><span class="mn" id="MathJax-Span-578" style="font-family: MathJax_Main; font-size: 70.7%;">2</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-579" style="font-family: MathJax_Main; padding-left: 0.222em;">×</span><span class="texatom" id="MathJax-Span-580" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-581"><span class="mo" id="MathJax-Span-582" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="mfrac" id="MathJax-Span-583" style="padding-left: 0.287em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-584" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.48em, 1000em, 2.473em, -0.418em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-585" style="font-family: MathJax_Main;">3</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-586" style="font-family: MathJax_Main; padding-left: 0.167em;">cos</span><span class="mo" id="MathJax-Span-587"></span><span class="mi" id="MathJax-Span-588" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span><span class="texatom" id="MathJax-Span-589"><span class="mrow" id="MathJax-Span-590"><span class="mo" id="MathJax-Span-591" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">)</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="mspace" id="MathJax-Span-592" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 0.167em;"></span><span class="mi" id="MathJax-Span-593" style="font-family: MathJax_Math; font-style: italic;">d<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-594" style="font-family: MathJax_Math; font-style: italic;">u</span></span></span><span style="display: inline-block; height: 3.268em; width: 0px;"></span></span><span style="clip: rect(1.754em, 1000em, 4.282em, -0.405em); left: 0em; position: absolute; top: -4.443em;"><span class="mtd" id="MathJax-Span-600"><span class="mrow" id="MathJax-Span-601"><span class="mo" id="MathJax-Span-602" style="font-family: MathJax_Size2; vertical-align: 0.001em;">∫<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.388em;"></span></span><span class="mfrac" id="MathJax-Span-603" style="padding-left: 0.287em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-604" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.48em, 1000em, 2.473em, -0.418em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-605" style="font-family: MathJax_Main;">3</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-606" style="padding-left: 0.167em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.76em;"><span style="clip: rect(1.697em, 1000em, 2.462em, -0.426em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-607" style="font-family: MathJax_Main;">cos</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 1.328em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-608" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-609"></span><span class="mi" id="MathJax-Span-610" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span><span class="mspace" id="MathJax-Span-611" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 0.167em;"></span><span class="mi" id="MathJax-Span-612" style="font-family: MathJax_Math; font-style: italic;">d<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-613" style="font-family: MathJax_Math; font-style: italic;">u</span></span></span><span style="display: inline-block; height: 3.268em; width: 0px;"></span></span><span style="clip: rect(1.754em, 1000em, 4.282em, -0.34em); left: 0em; position: absolute; top: -1.921em;"><span class="mtd" id="MathJax-Span-619"><span class="mrow" id="MathJax-Span-620"><span class="mfrac" id="MathJax-Span-621" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-622" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.48em, 1000em, 2.473em, -0.418em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-623" style="font-family: MathJax_Main;">3</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-624" style="font-family: MathJax_Size2; padding-left: 0.167em; vertical-align: 0.001em;">∫<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.388em;"></span></span><span class="mfrac" id="MathJax-Span-625" style="padding-left: 0.287em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-626" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-627" style="font-family: MathJax_Main;">2</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-628" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mn" id="MathJax-Span-629" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-630" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-631" style="font-family: MathJax_Main; padding-left: 0.222em;">cos</span><span class="mo" id="MathJax-Span-632"></span><span class="mn" id="MathJax-Span-633" style="font-family: MathJax_Main; padding-left: 0.167em;">2</span><span class="mi" id="MathJax-Span-634" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mo" id="MathJax-Span-635" style="font-family: MathJax_Main;">)</span><span class="mspace" id="MathJax-Span-636" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 0.167em;"></span><span class="mi" id="MathJax-Span-637" style="font-family: MathJax_Math; font-style: italic;">d<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-638" style="font-family: MathJax_Math; font-style: italic;">u</span></span></span><span style="display: inline-block; height: 3.268em; width: 0px;"></span></span><span style="clip: rect(0.802em, 1000em, 3.159em, -0.34em); left: 0em; position: absolute; top: 1.552em;"><span class="mtd" id="MathJax-Span-644"><span class="mrow" id="MathJax-Span-645"><span class="mfrac" id="MathJax-Span-646" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-647" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.473em, -0.418em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-648" style="font-family: MathJax_Main;">6</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-649" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mi" id="MathJax-Span-650" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mo" id="MathJax-Span-651" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-652" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-653" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-654" style="font-family: MathJax_Main;">2</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-655" style="font-family: MathJax_Main; padding-left: 0.167em;">sin</span><span class="mo" id="MathJax-Span-656"></span><span class="mn" id="MathJax-Span-657" style="font-family: MathJax_Main; padding-left: 0.167em;">2</span><span class="mi" id="MathJax-Span-658" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mo" id="MathJax-Span-659" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-660" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-661" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">k</span><span class="mo" id="MathJax-Span-662" style="font-family: MathJax_Main;">.</span><span class="mo" id="MathJax-Span-663" style="font-family: MathJax_Main; padding-left: 0.167em;">.</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="display: inline-block; height: 10.621em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 13.217em; overflow: hidden; vertical-align: -6.269em; width: 0px;"></span></span></nobr></span></div><br /><br />
To return to an expression in terms of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-58-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-664"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-665"><span class="mi" id="MathJax-Span-666" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> we use <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-59-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-667"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 324px;"><span style="clip: rect(1.137em, 1000em, 2.701em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-668"><span class="mi" id="MathJax-Span-669" style="font-family: MathJax_Main;">sin</span><span class="mo" id="MathJax-Span-670"></span><span class="mn" id="MathJax-Span-671" style="font-family: MathJax_Main; padding-left: 0.167em;">2</span><span class="mi" id="MathJax-Span-672" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mo" id="MathJax-Span-673" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-674" style="font-family: MathJax_Main; padding-left: 0.278em;">2</span><span class="mi" id="MathJax-Span-675" style="font-family: MathJax_Main; padding-left: 0.167em;">sin</span><span class="mo" id="MathJax-Span-676"></span><span class="mi" id="MathJax-Span-677" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span><span class="mspace" id="MathJax-Span-678" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 0.167em;"></span><span class="mi" id="MathJax-Span-679" style="font-family: MathJax_Main; padding-left: 0.167em;">cos</span><span class="mo" id="MathJax-Span-680"></span><span class="mi" id="MathJax-Span-681" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">u</span><span class="mo" id="MathJax-Span-682" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-683" style="font-family: MathJax_Main; padding-left: 0.278em;">6</span><span class="mi" id="MathJax-Span-684" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-685" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-686" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-687" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-688" style="font-family: MathJax_Main; padding-left: 0.222em;">9</span><span class="msubsup" id="MathJax-Span-689"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-690" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-691" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-692"><span style="display: inline-block; height: 0px; position: relative; width: 1.556em;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.405em); left: 0em; position: absolute; top: -2.298em;"><span class="mo" id="MathJax-Span-693" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.724em;"><span class="texatom" id="MathJax-Span-694"><span class="mrow" id="MathJax-Span-695"><span class="mn" id="MathJax-Span-696" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span class="texatom" id="MathJax-Span-697"><span class="mrow" id="MathJax-Span-698"><span class="mo" id="MathJax-Span-699" style="font-family: MathJax_Main; font-size: 70.7%;">/</span></span></span><span class="mn" id="MathJax-Span-700" style="font-family: MathJax_Main; font-size: 70.7%;">2</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.849em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>, and the integral we want is<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-60-Frame"><nobr><span class="math" id="MathJax-Span-701"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 325px;"><span style="clip: rect(0.904em, 1000em, 3.261em, -0.434em); left: 0em; position: absolute; top: -2.4em;"><span class="mrow" id="MathJax-Span-702"><span class="mi" id="MathJax-Span-703" style="font-family: MathJax_Math; font-style: italic;">I<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.064em;"></span></span><span class="mo" id="MathJax-Span-704" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mfrac" id="MathJax-Span-705" style="padding-left: 0.398em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-706" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.473em, -0.418em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-707" style="font-family: MathJax_Main;">6</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-708" style="padding-left: 0.167em;"><span style="display: inline-block; height: 0px; position: relative; width: 2.22em;"><span style="clip: rect(1.476em, 1000em, 2.462em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-709" style="font-family: MathJax_Main;">sin</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 1.225em; position: absolute; top: -2.66em;"><span class="texatom" id="MathJax-Span-710"><span class="mrow" id="MathJax-Span-711"><span class="mo" id="MathJax-Span-712" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-713" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-714"></span><span class="mn" id="MathJax-Span-715" style="font-family: MathJax_Main; padding-left: 0.167em;">3</span><span class="mi" id="MathJax-Span-716" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-717" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-718" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.631em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-719" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 50%; margin-left: -0.255em; position: absolute; top: -1.612em;"><span class="mn" id="MathJax-Span-720" style="font-family: MathJax_Main;">2</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.631em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-721" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">x</span><span class="texatom" id="MathJax-Span-722"><span class="mrow" id="MathJax-Span-723"><span class="mo" id="MathJax-Span-724" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="mn" id="MathJax-Span-725" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-726" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-727" style="font-family: MathJax_Main; padding-left: 0.222em;">9</span><span class="msubsup" id="MathJax-Span-728"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-729" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-730" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-731"><span style="display: inline-block; height: 0px; position: relative; width: 1.607em;"><span style="clip: rect(1.397em, 1000em, 2.903em, -0.425em); left: 0em; position: absolute; top: -2.4em;"><span class="texatom" id="MathJax-Span-732"><span class="mrow" id="MathJax-Span-733"><span class="mo" id="MathJax-Span-734" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">)</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span style="display: inline-block; height: 2.4em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.823em;"><span class="texatom" id="MathJax-Span-735"><span class="mrow" id="MathJax-Span-736"><span class="mn" id="MathJax-Span-737" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span class="texatom" id="MathJax-Span-738"><span class="mrow" id="MathJax-Span-739"><span class="mo" id="MathJax-Span-740" style="font-family: MathJax_Main; font-size: 70.7%;">/</span></span></span><span class="mn" id="MathJax-Span-741" style="font-family: MathJax_Main; font-size: 70.7%;">2</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-742" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-743" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">k</span><span class="mo" id="MathJax-Span-744" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; height: 2.4em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 2.928em; overflow: hidden; vertical-align: -1.032em; width: 0px;"></span></span></nobr></span></div><br /><br />
<br /><br />
<span style="font-weight: bold;">Example <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-61-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-745"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.468em, 1000em, 2.451em, -0.432em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-746"><span class="mn" id="MathJax-Span-747" style="font-family: MathJax_Main;">4</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.06em; overflow: hidden; vertical-align: -0.069em; width: 0px;"></span></span></nobr></span> Area inside an ellips</span>e<br /><br />
<img alt="ellipse" height="173" src="https://nrich.maths.org/content/id/7075/ellipse.jpg" width="600" /><br /><br />
In order to find the the area inside the ellipse <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-62-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-748"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 102px;"><span style="clip: rect(1.004em, 1000em, 2.995em, -0.34em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-749"><span class="mfrac" id="MathJax-Span-750" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.837em;"><span style="clip: rect(3.24em, 1000em, 4.144em, -0.435em); left: 50%; margin-left: -0.358em; position: absolute; top: -4.396em;"><span class="msubsup" id="MathJax-Span-751"><span style="display: inline-block; height: 0px; position: relative; width: 0.717em;"><span style="clip: rect(1.781em, 1000em, 2.408em, -0.435em); left: 0em; position: absolute; top: -2.247em;"><span class="mi" id="MathJax-Span-752" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.503em;"><span class="mn" id="MathJax-Span-753" style="font-family: MathJax_Main; font-size: 50%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(3.292em, 1000em, 4.143em, -0.436em); left: 50%; margin-left: -0.358em; position: absolute; top: -3.54em;"><span class="msubsup" id="MathJax-Span-754"><span style="display: inline-block; height: 0px; position: relative; width: 0.717em;"><span style="clip: rect(1.782em, 1000em, 2.407em, -0.436em); left: 0em; position: absolute; top: -2.247em;"><span class="mi" id="MathJax-Span-755" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">a</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.451em;"><span class="mn" id="MathJax-Span-756" style="font-family: MathJax_Main; font-size: 50%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.837em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-757" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mfrac" id="MathJax-Span-758" style="padding-left: 0.342em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.837em;"><span style="clip: rect(3.24em, 1000em, 4.281em, -0.445em); left: 50%; margin-left: -0.359em; position: absolute; top: -4.534em;"><span class="msubsup" id="MathJax-Span-759"><span style="display: inline-block; height: 0px; position: relative; width: 0.717em;"><span style="clip: rect(1.781em, 1000em, 2.545em, -0.445em); left: 0em; position: absolute; top: -2.247em;"><span class="mi" id="MathJax-Span-760" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.004em;"></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="left: 0.409em; position: absolute; top: -2.503em;"><span class="mn" id="MathJax-Span-761" style="font-family: MathJax_Main; font-size: 50%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(3.199em, 1000em, 4.144em, -0.431em); left: 50%; margin-left: -0.307em; position: absolute; top: -3.446em;"><span class="msubsup" id="MathJax-Span-762"><span style="display: inline-block; height: 0px; position: relative; width: 0.615em;"><span style="clip: rect(1.603em, 1000em, 2.408em, -0.431em); left: 0em; position: absolute; top: -2.247em;"><span class="mi" id="MathJax-Span-763" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">b</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="left: 0.306em; position: absolute; top: -2.544em;"><span class="mn" id="MathJax-Span-764" style="font-family: MathJax_Main; font-size: 50%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.837em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-765" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-766" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 2.43em; overflow: hidden; vertical-align: -0.81em; width: 0px;"></span></span></nobr></span>, we can use the transformation <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-63-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-767"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 135px;"><span style="clip: rect(1.24em, 1000em, 2.803em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-768"><span class="mo" id="MathJax-Span-769" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-770" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-771" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-772" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-773" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-774" style="font-family: MathJax_Main; padding-left: 0.278em;">→</span><span class="mo" id="MathJax-Span-775" style="font-family: MathJax_Main; padding-left: 0.278em;">(</span><span class="mfrac" id="MathJax-Span-776" style="padding-left: 0.12em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.835em;"><span style="clip: rect(1.603em, 1000em, 2.408em, -0.431em); left: 50%; margin-left: -0.357em; position: absolute; top: -2.66em;"><span class="mrow" id="MathJax-Span-777"><span class="mi" id="MathJax-Span-778" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">b</span><span class="mi" id="MathJax-Span-779" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(1.782em, 1000em, 2.407em, -0.436em); left: 50%; margin-left: -0.204em; position: absolute; top: -1.902em;"><span class="mi" id="MathJax-Span-780" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">a</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.835em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-781" style="font-family: MathJax_Main; padding-left: 0.167em;">,</span><span class="mi" id="MathJax-Span-782" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-783" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.847em; overflow: hidden; vertical-align: -0.548em; width: 0px;"></span></span></nobr></span> to change the ellipse into a circle. Since the lengths in the <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-64-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-784"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-785"><span class="mi" id="MathJax-Span-786" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>--direction are changed by a factor <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-65-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-787"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 29px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.42em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-788"><span class="mi" id="MathJax-Span-789" style="font-family: MathJax_Math; font-style: italic;">b</span><span class="texatom" id="MathJax-Span-790"><span class="mrow" id="MathJax-Span-791"><span class="mo" id="MathJax-Span-792" style="font-family: MathJax_Main;">/</span></span></span><span class="mi" id="MathJax-Span-793" style="font-family: MathJax_Math; font-style: italic;">a</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>, and the lengths in the <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-66-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-794"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.703em, 1000em, 2.656em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-795"><span class="mi" id="MathJax-Span-796" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.019em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span>--direction remain the same, the area is changed by a factor <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-67-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-797"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 29px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.42em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-798"><span class="mi" id="MathJax-Span-799" style="font-family: MathJax_Math; font-style: italic;">b</span><span class="texatom" id="MathJax-Span-800"><span class="mrow" id="MathJax-Span-801"><span class="mo" id="MathJax-Span-802" style="font-family: MathJax_Main;">/</span></span></span><span class="mi" id="MathJax-Span-803" style="font-family: MathJax_Math; font-style: italic;">a</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>. Thus<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-68-Frame"><nobr><span class="math" id="MathJax-Span-804"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 301px;"><span style="clip: rect(0.774em, 1000em, 3.147em, -0.46em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-805"><span class="mtext" id="MathJax-Span-806" style="font-family: MathJax_Main;">Area of circle</span><span class="mo" id="MathJax-Span-807" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mfrac" id="MathJax-Span-808" style="padding-left: 0.398em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.682em;"><span style="clip: rect(1.451em, 1000em, 2.462em, -0.42em); left: 50%; margin-left: -0.204em; position: absolute; top: -2.974em;"><span class="mi" id="MathJax-Span-809" style="font-family: MathJax_Math; font-style: italic;">b</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 50%; margin-left: -0.281em; position: absolute; top: -1.612em;"><span class="mi" id="MathJax-Span-810" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 0.682em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-811" style="font-family: MathJax_Main; padding-left: 0.222em;">×</span><span class="mtext" id="MathJax-Span-812" style="font-family: MathJax_Main; padding-left: 0.222em;">Area of ellipse</span><span class="mo" id="MathJax-Span-813" style="font-family: MathJax_Main;">,</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 2.949em; overflow: hidden; vertical-align: -1.016em; width: 0px;"></span></span></nobr></span></div><br /><br />
which gives the area of the ellipse as <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-69-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-814"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 97px;"><span style="clip: rect(1.253em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-815"><span class="mo" id="MathJax-Span-816" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-817" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="texatom" id="MathJax-Span-818"><span class="mrow" id="MathJax-Span-819"><span class="mo" id="MathJax-Span-820" style="font-family: MathJax_Main;">/</span></span></span><span class="mi" id="MathJax-Span-821" style="font-family: MathJax_Math; font-style: italic;">b</span><span class="mo" id="MathJax-Span-822" style="font-family: MathJax_Main; padding-left: 0.222em;">×</span><span class="mi" id="MathJax-Span-823" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">π<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-824"><span style="display: inline-block; height: 0px; position: relative; width: 0.841em;"><span style="clip: rect(1.451em, 1000em, 2.462em, -0.42em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-825" style="font-family: MathJax_Math; font-style: italic;">b</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.668em;"><span class="mn" id="MathJax-Span-826" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-827" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.692em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>, that is <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-70-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-828"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 31px;"><span style="clip: rect(1.451em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-829"><span class="mi" id="MathJax-Span-830" style="font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-831" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="mi" id="MathJax-Span-832" style="font-family: MathJax_Math; font-style: italic;">b</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.098em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>.<br /><br />
<span style="font-weight: bold;"> </span><br /><br />
<span style="font-weight: bold;">Example <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-71-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-833"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.41em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-834"><span class="mn" id="MathJax-Span-835" style="font-family: MathJax_Main;">5</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> Polynomial</span> <br /><br />
<br /><br />
Consider a general polynomial <span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-72-Frame"><nobr><span class="math" id="MathJax-Span-836"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 422px;"><span style="clip: rect(1.261em, 1000em, 2.701em, -0.499em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-837"><span class="mi" id="MathJax-Span-838" style="font-family: MathJax_Math; font-style: italic;">p</span><span class="mo" id="MathJax-Span-839" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-840" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-841" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-842" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="msubsup" id="MathJax-Span-843" style="padding-left: 0.278em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.994em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-844" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mn" id="MathJax-Span-845" style="font-family: MathJax_Main; font-size: 70.7%;">0</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-846" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-847" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.994em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-848" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mn" id="MathJax-Span-849" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-850" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-851" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-852" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.994em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-853" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mn" id="MathJax-Span-854" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-855"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-856" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-857" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-858" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-859" style="font-family: MathJax_Main; padding-left: 0.222em;">⋯</span><span class="mo" id="MathJax-Span-860" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-861" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.964em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-862" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="texatom" id="MathJax-Span-863"><span class="mrow" id="MathJax-Span-864"><span class="mi" id="MathJax-Span-865" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-866" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-867" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-868"><span style="display: inline-block; height: 0px; position: relative; width: 2.015em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-869" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="texatom" id="MathJax-Span-870"><span class="mrow" id="MathJax-Span-871"><span class="mi" id="MathJax-Span-872" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-873" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-874" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-875" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-876" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-877" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mi" id="MathJax-Span-878" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-879"><span style="display: inline-block; height: 0px; position: relative; width: 1.096em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-880" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mi" id="MathJax-Span-881" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-882" style="font-family: MathJax_Main;">.</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.681em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div>Let us make the substitution <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-73-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-883"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 79px;"><span style="clip: rect(1.451em, 1000em, 2.533em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-884"><span class="mi" id="MathJax-Span-885" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-886" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-887" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">t</span><span class="mo" id="MathJax-Span-888" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-889" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">k</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.194em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span>, where <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-74-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-890"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.451em, 1000em, 2.462em, -0.412em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-891"><span class="mi" id="MathJax-Span-892" style="font-family: MathJax_Math; font-style: italic;">k</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.098em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> is a constant which we shall determine later. Now write <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-75-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-893"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 186px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.499em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-894"><span class="mi" id="MathJax-Span-895" style="font-family: MathJax_Math; font-style: italic;">p</span><span class="mo" id="MathJax-Span-896" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-897" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-898" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-899" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-900" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">p</span><span class="mo" id="MathJax-Span-901" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-902" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-903" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-904" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">k</span><span class="mo" id="MathJax-Span-905" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-906" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-907" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">q<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.014em;"></span></span><span class="mo" id="MathJax-Span-908" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-909" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-910" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>. Then<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-76-Frame"><nobr><span class="math" id="MathJax-Span-911"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 486px;"><span style="clip: rect(-0.289em, 1000em, 4.282em, -0.293em); left: 0em; position: absolute; top: -2.247em;"><span class="mrow" id="MathJax-Span-912"><span class="mtable" id="MathJax-Span-913" style="padding-left: 0.167em; padding-right: 0.167em;"><span style="display: inline-block; height: 0px; position: relative; width: 24.504em;"><span style="clip: rect(2.36em, 1000em, 6.384em, -0.46em); left: 0em; position: absolute; top: -4.698em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.583em;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.427em); position: absolute; right: 0em; top: -3.732em;"><span class="mtd" id="MathJax-Span-914"><span class="mrow" id="MathJax-Span-915"><span class="mi" id="MathJax-Span-916" style="font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.014em;"></span></span><span class="mo" id="MathJax-Span-917" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-918" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-919" style="font-family: MathJax_Main;">)</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(3.83em, 1000em, 4.136em, -0.46em); position: absolute; right: 0em; top: -3.983em;"><span class="mtd" id="MathJax-Span-969"><span class="mrow" id="MathJax-Span-970"></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(3.83em, 1000em, 4.136em, -0.46em); position: absolute; right: 0em; top: -2.45em;"><span class="mtd" id="MathJax-Span-1018"><span class="mrow" id="MathJax-Span-1019"></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span><span style="display: inline-block; height: 4.698em; width: 0px;"></span></span><span style="clip: rect(2.335em, 1000em, 5.942em, -0.404em); left: 1.861em; position: absolute; top: -4.289em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.766em;"><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: -3.732em;"><span class="mtd" id="MathJax-Span-920"><span class="mrow" id="MathJax-Span-921"><span class="mo" id="MathJax-Span-922" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: -2.298em;"><span class="mtd" id="MathJax-Span-971"><span class="mrow" id="MathJax-Span-972"><span class="mo" id="MathJax-Span-973" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.778em, 1000em, 2.418em, -0.404em); left: 50%; margin-left: -0.383em; position: absolute; top: -0.765em;"><span class="mtd" id="MathJax-Span-1020"><span class="mrow" id="MathJax-Span-1021"><span class="mo" id="MathJax-Span-1022" style="font-family: MathJax_Main;">=</span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="display: inline-block; height: 4.289em; width: 0px;"></span></span><span style="clip: rect(2.724em, 1000em, 7.295em, -0.427em); left: 2.904em; position: absolute; top: -5.259em;"><span style="display: inline-block; height: 0px; position: relative; width: 21.599em;"><span style="clip: rect(1.197em, 1000em, 2.701em, -0.427em); left: 0em; position: absolute; top: -3.732em;"><span class="mtd" id="MathJax-Span-923"><span class="mrow" id="MathJax-Span-924"><span class="msubsup" id="MathJax-Span-925"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-926" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mi" id="MathJax-Span-927" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-928" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-929" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-930" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-931" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">k</span><span class="msubsup" id="MathJax-Span-932"><span style="display: inline-block; height: 0px; position: relative; width: 0.892em;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.405em); left: 0em; position: absolute; top: -2.298em;"><span class="mo" id="MathJax-Span-933" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.724em;"><span class="mi" id="MathJax-Span-934" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-935" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-936" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.964em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-937" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="texatom" id="MathJax-Span-938"><span class="mrow" id="MathJax-Span-939"><span class="mi" id="MathJax-Span-940" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-941" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-942" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-943" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-944" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-945" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-946" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">k</span><span class="msubsup" id="MathJax-Span-947"><span style="display: inline-block; height: 0px; position: relative; width: 1.811em;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.405em); left: 0em; position: absolute; top: -2.298em;"><span class="mo" id="MathJax-Span-948" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.724em;"><span class="texatom" id="MathJax-Span-949"><span class="mrow" id="MathJax-Span-950"><span class="mi" id="MathJax-Span-951" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-952" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-953" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-954" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-955" style="font-family: MathJax_Main; padding-left: 0.222em;">⋯</span><span class="mo" id="MathJax-Span-956" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-957" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.994em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-958" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mn" id="MathJax-Span-959" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-960" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-961" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-962" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-963" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">k</span><span class="mo" id="MathJax-Span-964" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-965" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-966" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.994em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-967" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mn" id="MathJax-Span-968" style="font-family: MathJax_Main; font-size: 70.7%;">0</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(1.363em, 1000em, 2.903em, -0.308em); left: 0em; position: absolute; top: -2.4em;"><span class="mtd" id="MathJax-Span-974"><span class="mrow" id="MathJax-Span-975"><span class="texatom" id="MathJax-Span-976"><span class="mrow" id="MathJax-Span-977"><span class="mo" id="MathJax-Span-978" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="msubsup" id="MathJax-Span-979"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-980" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mi" id="MathJax-Span-981" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-982"><span style="display: inline-block; height: 0px; position: relative; width: 0.841em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-983" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.66em;"><span class="mi" id="MathJax-Span-984" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-985" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-986" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">n</span><span class="msubsup" id="MathJax-Span-987"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-988" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mi" id="MathJax-Span-989" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-990" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="msubsup" id="MathJax-Span-991"><span style="display: inline-block; height: 0px; position: relative; width: 1.76em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-992" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.66em;"><span class="texatom" id="MathJax-Span-993"><span class="mrow" id="MathJax-Span-994"><span class="mi" id="MathJax-Span-995" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-996" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-997" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-998" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-999" style="font-family: MathJax_Main; padding-left: 0.222em;">⋯</span><span class="mo" id="MathJax-Span-1000" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-1001" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-1002" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.964em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1003" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="texatom" id="MathJax-Span-1004"><span class="mrow" id="MathJax-Span-1005"><span class="mi" id="MathJax-Span-1006" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1007" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1008" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1009"><span style="display: inline-block; height: 0px; position: relative; width: 1.76em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1010" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.66em;"><span class="texatom" id="MathJax-Span-1011"><span class="mrow" id="MathJax-Span-1012"><span class="mi" id="MathJax-Span-1013" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1014" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1015" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1016" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-1017" style="font-family: MathJax_Main; padding-left: 0.222em;">⋯</span></span></span><span style="display: inline-block; height: 2.4em; width: 0px;"></span></span><span style="clip: rect(1.363em, 1000em, 2.903em, -0.427em); left: 0em; position: absolute; top: -0.867em;"><span class="mtd" id="MathJax-Span-1023"><span class="mrow" id="MathJax-Span-1024"><span class="msubsup" id="MathJax-Span-1025"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1026" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mi" id="MathJax-Span-1027" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1028"><span style="display: inline-block; height: 0px; position: relative; width: 0.841em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1029" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.66em;"><span class="mi" id="MathJax-Span-1030" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1031" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="texatom" id="MathJax-Span-1032" style="padding-left: 0.222em;"><span class="mrow" id="MathJax-Span-1033"><span class="mo" id="MathJax-Span-1034" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">(</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="mi" id="MathJax-Span-1035" style="font-family: MathJax_Math; font-style: italic;">n</span><span class="msubsup" id="MathJax-Span-1036"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1037" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mi" id="MathJax-Span-1038" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1039" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-1040" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-1041" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.964em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1042" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="texatom" id="MathJax-Span-1043"><span class="mrow" id="MathJax-Span-1044"><span class="mi" id="MathJax-Span-1045" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1046" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1047" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-1048"><span class="mrow" id="MathJax-Span-1049"><span class="mo" id="MathJax-Span-1050" style="vertical-align: 0em;"><span style="font-family: MathJax_Size1;">)</span></span><span style="border-left: 0em solid; display: inline-block; height: 0.849em; overflow: hidden; vertical-align: 0em; width: 0px;"></span></span></span><span class="msubsup" id="MathJax-Span-1051"><span style="display: inline-block; height: 0px; position: relative; width: 1.76em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1052" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.66em;"><span class="texatom" id="MathJax-Span-1053"><span class="mrow" id="MathJax-Span-1054"><span class="mi" id="MathJax-Span-1055" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1056" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1057" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1058" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-1059" style="font-family: MathJax_Main; padding-left: 0.222em;">⋯</span><span class="mo" id="MathJax-Span-1060" style="font-family: MathJax_Main; padding-left: 0.167em;">,</span></span></span><span style="display: inline-block; height: 2.4em; width: 0px;"></span></span></span><span style="display: inline-block; height: 5.259em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 5.939em; overflow: hidden; vertical-align: -2.629em; width: 0px;"></span></span></nobr></span></div><br /><br />
where here "<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-77-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1061"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 23px;"><span style="clip: rect(1.835em, 1000em, 2.261em, -0.382em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1062"><span class="mo" id="MathJax-Span-1063" style="font-family: MathJax_Main;">⋯</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.302em; overflow: hidden; vertical-align: 0.189em; width: 0px;"></span></span></nobr></span>'' means powers of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-78-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1064"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 7px;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1065"><span class="mi" id="MathJax-Span-1066" style="font-family: MathJax_Math; font-style: italic;">t</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.005em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> of order <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-79-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1067"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 46px;"><span style="clip: rect(1.479em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1068"><span class="mi" id="MathJax-Span-1069" style="font-family: MathJax_Math; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1070" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-1071" style="font-family: MathJax_Main; padding-left: 0.222em;">2</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.06em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> or less. If we now choose <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-80-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1072"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 132px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.412em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1073"><span class="mi" id="MathJax-Span-1074" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-1075" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mo" id="MathJax-Span-1076" style="font-family: MathJax_Main; padding-left: 0.278em;">−</span><span class="msubsup" id="MathJax-Span-1077"><span style="display: inline-block; height: 0px; position: relative; width: 1.964em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1078" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="texatom" id="MathJax-Span-1079"><span class="mrow" id="MathJax-Span-1080"><span class="mi" id="MathJax-Span-1081" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1082" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1083" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-1084"><span class="mrow" id="MathJax-Span-1085"><span class="mo" id="MathJax-Span-1086" style="font-family: MathJax_Main;">/</span></span></span><span class="mi" id="MathJax-Span-1087" style="font-family: MathJax_Math; font-style: italic;">n</span><span class="msubsup" id="MathJax-Span-1088"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1089" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mi" id="MathJax-Span-1090" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span> we see that<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-81-Frame"><nobr><span class="math" id="MathJax-Span-1091"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 328px;"><span style="clip: rect(1.261em, 1000em, 2.701em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1092"><span class="mi" id="MathJax-Span-1093" style="font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.014em;"></span></span><span class="mo" id="MathJax-Span-1094" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-1095" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-1096" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-1097" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="msubsup" id="MathJax-Span-1098" style="padding-left: 0.278em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.704em, 1000em, 2.461em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1099" style="font-family: MathJax_Math; font-style: italic;">a</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.097em;"><span class="mi" id="MathJax-Span-1100" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1101"><span style="display: inline-block; height: 0px; position: relative; width: 0.841em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1102" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.66em;"><span class="mi" id="MathJax-Span-1103" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1104" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-1105" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.811em;"><span style="clip: rect(1.451em, 1000em, 2.462em, -0.42em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1106" style="font-family: MathJax_Math; font-style: italic;">b</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.097em;"><span class="texatom" id="MathJax-Span-1107"><span class="mrow" id="MathJax-Span-1108"><span class="mi" id="MathJax-Span-1109" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1110" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1111" style="font-family: MathJax_Main; font-size: 70.7%;">2</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1112"><span style="display: inline-block; height: 0px; position: relative; width: 1.76em;"><span style="clip: rect(1.519em, 1000em, 2.462em, -0.441em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1113" style="font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.357em; position: absolute; top: -2.66em;"><span class="texatom" id="MathJax-Span-1114"><span class="mrow" id="MathJax-Span-1115"><span class="mi" id="MathJax-Span-1116" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1117" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1118" style="font-family: MathJax_Main; font-size: 70.7%;">2</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1119" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-1120" style="font-family: MathJax_Main; padding-left: 0.222em;">⋯</span><span class="mo" id="MathJax-Span-1121" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-1122" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.841em;"><span style="clip: rect(1.451em, 1000em, 2.462em, -0.42em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1123" style="font-family: MathJax_Math; font-style: italic;">b</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.097em;"><span class="mn" id="MathJax-Span-1124" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-1125" style="font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-1126" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-1127" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.841em;"><span style="clip: rect(1.451em, 1000em, 2.462em, -0.42em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1128" style="font-family: MathJax_Math; font-style: italic;">b</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.408em; position: absolute; top: -2.097em;"><span class="mn" id="MathJax-Span-1129" style="font-family: MathJax_Main; font-size: 70.7%;">0</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1130" style="font-family: MathJax_Main;">;</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.681em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div><br /><br />
in other words, by changing the variable we can remove the term of degree <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-82-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1131"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 46px;"><span style="clip: rect(1.479em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1132"><span class="mi" id="MathJax-Span-1133" style="font-family: MathJax_Math; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1134" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" id="MathJax-Span-1135" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.06em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>. While the effect of this substitution may not seem spectacular, it is important. It is <span style="font-style: italic;">exactly</span><br />
what we do when we 'complete the square' to solve quadratic equations,<br />
and this is the method used to find the formula for the roots of a <br />
quadratic equation. It is also the<br />
first step in solving cubic equations, for there it says that we only <br />
need consider equations of the form <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-83-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1136"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 133px;"><span style="clip: rect(1.311em, 1000em, 2.533em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1137"><span class="msubsup" id="MathJax-Span-1138"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1139" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="mn" id="MathJax-Span-1140" style="font-family: MathJax_Main; font-size: 70.7%;">3</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1141" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1142" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">b</span><span class="mi" id="MathJax-Span-1143" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-1144" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1145" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">c</span><span class="mo" id="MathJax-Span-1146" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-1147" style="font-family: MathJax_Main; padding-left: 0.278em;">0</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.383em; overflow: hidden; vertical-align: -0.181em; width: 0px;"></span></span></nobr></span>.<br /><br />
<br /><br />
Finally, it is worth noting that the coefficient of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-84-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1148"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 39px;"><span style="clip: rect(1.26em, 1000em, 2.411em, -0.425em); left: 0em; position: absolute; top: -2.247em;"><span class="mrow" id="MathJax-Span-1149"><span class="msubsup" id="MathJax-Span-1150"><span style="display: inline-block; height: 0px; position: relative; width: 2.015em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1151" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="texatom" id="MathJax-Span-1152"><span class="mrow" id="MathJax-Span-1153"><span class="mi" id="MathJax-Span-1154" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1155" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1156" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.288em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> in a polynomial equation of degree <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-85-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1157"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1158"><span class="mi" id="MathJax-Span-1159" style="font-family: MathJax_Math; font-style: italic;">n</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> is minus the sum of the roots of the equation so this substitution is such that the chosen value of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-86-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1160"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.451em, 1000em, 2.462em, -0.412em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1161"><span class="mi" id="MathJax-Span-1162" style="font-family: MathJax_Math; font-style: italic;">k</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.098em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> is the average value of the roots of the polynomial.<br /><br />
<br /><br />
<span style="font-weight: bold;">Example <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-87-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1163"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.418em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1164"><span class="mn" id="MathJax-Span-1165" style="font-family: MathJax_Main;">6</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> Transformations of the plane</span> <br /><br />
In Example <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-88-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1166"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.479em, 1000em, 2.473em, -0.41em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1167"><span class="mn" id="MathJax-Span-1168" style="font-family: MathJax_Main;">5</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.075em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span> we showed how to remove the term in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-89-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1169"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 39px;"><span style="clip: rect(1.26em, 1000em, 2.411em, -0.425em); left: 0em; position: absolute; top: -2.247em;"><span class="mrow" id="MathJax-Span-1170"><span class="msubsup" id="MathJax-Span-1171"><span style="display: inline-block; height: 0px; position: relative; width: 2.015em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1172" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.61em;"><span class="texatom" id="MathJax-Span-1173"><span class="mrow" id="MathJax-Span-1174"><span class="mi" id="MathJax-Span-1175" style="font-family: MathJax_Math; font-size: 70.7%; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1176" style="font-family: MathJax_Main; font-size: 70.7%;">−</span><span class="mn" id="MathJax-Span-1177" style="font-family: MathJax_Main; font-size: 70.7%;">1</span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.288em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> from a polynomial of degree <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-90-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1178"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1179"><span class="mi" id="MathJax-Span-1180" style="font-family: MathJax_Math; font-style: italic;">n</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span>. Now we are going to show how, given the equation of a conic, for example,<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-91-Frame"><nobr><span class="math" id="MathJax-Span-1181"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 218px;"><span style="clip: rect(1.261em, 1000em, 2.701em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1182"><span class="msubsup" id="MathJax-Span-1183"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1184" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-1185" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1186" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" id="MathJax-Span-1187" style="font-family: MathJax_Main; padding-left: 0.222em;">2</span><span class="mi" id="MathJax-Span-1188" style="font-family: MathJax_Math; font-style: italic;">b</span><span class="mi" id="MathJax-Span-1189" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mi" id="MathJax-Span-1190" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-1191" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-1192" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.995em;"><span style="clip: rect(1.703em, 1000em, 2.656em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1193" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-1194" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1195" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-1196" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span><span class="mo" id="MathJax-Span-1197" style="font-family: MathJax_Main;">,</span><span class="mspace" id="MathJax-Span-1198" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 1em;"></span><span class="mo" id="MathJax-Span-1199" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mn" id="MathJax-Span-1200" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-1201" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.681em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div><br /><br />
we can remove the <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-92-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1202"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 22px;"><span style="clip: rect(1.703em, 1000em, 2.656em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1203"><span class="mi" id="MathJax-Span-1204" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mi" id="MathJax-Span-1205" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.019em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span> term and so more easily discover the properties of the conic. First, if we make the substitution<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-93-Frame"><nobr><span class="math" id="MathJax-Span-1206"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 349px;"><span style="clip: rect(0.802em, 1000em, 3.419em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1207"><span class="mi" id="MathJax-Span-1208" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-1209" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mfrac" id="MathJax-Span-1210" style="padding-left: 0.398em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.448em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-1211" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(2.985em, 1000em, 4.291em, -0.388em); left: 50%; margin-left: -0.664em; position: absolute; top: -3.17em;"><span class="msqrt" id="MathJax-Span-1212"><span style="display: inline-block; height: 0px; position: relative; width: 1.328em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 0.817em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1213"><span class="mn" id="MathJax-Span-1214" style="font-family: MathJax_Main;">2</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.919em, 1000em, 1.289em, -0.46em); left: 0.817em; position: absolute; top: -1.917em;"><span style="border-left: 0.511em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: -0.064em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span><span style="clip: rect(3.03em, 1000em, 4.336em, -0.388em); left: 0em; position: absolute; top: -4.028em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.448em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1215" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mi" id="MathJax-Span-1216" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mo" id="MathJax-Span-1217" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-1218" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">v</span><span class="mo" id="MathJax-Span-1219" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-1220" style="font-family: MathJax_Main;">,</span><span class="mspace" id="MathJax-Span-1221" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 1em;"></span><span class="mi" id="MathJax-Span-1222" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-1223" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mfrac" id="MathJax-Span-1224" style="padding-left: 0.398em; padding-right: 0.12em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.448em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.377em); left: 50%; margin-left: -0.255em; position: absolute; top: -2.974em;"><span class="mn" id="MathJax-Span-1225" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(2.985em, 1000em, 4.291em, -0.388em); left: 50%; margin-left: -0.664em; position: absolute; top: -3.17em;"><span class="msqrt" id="MathJax-Span-1226"><span style="display: inline-block; height: 0px; position: relative; width: 1.328em;"><span style="clip: rect(1.479em, 1000em, 2.451em, -0.41em); left: 0.817em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1227"><span class="mn" id="MathJax-Span-1228" style="font-family: MathJax_Main;">2</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="clip: rect(0.919em, 1000em, 1.289em, -0.46em); left: 0.817em; position: absolute; top: -1.917em;"><span style="border-left: 0.511em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: -0.064em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span><span style="clip: rect(3.03em, 1000em, 4.336em, -0.388em); left: 0em; position: absolute; top: -4.028em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span></span></span><span style="display: inline-block; height: 3.983em; width: 0px;"></span></span><span style="clip: rect(0.855em, 1000em, 1.225em, -0.46em); left: 0em; position: absolute; top: -1.292em;"><span style="border-left: 1.448em solid; display: inline-block; height: 1.25px; overflow: hidden; vertical-align: 0em; width: 0px;"></span><span style="display: inline-block; height: 1.072em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1229" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mi" id="MathJax-Span-1230" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mo" id="MathJax-Span-1231" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1232" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">v</span><span class="mo" id="MathJax-Span-1233" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-1234" style="font-family: MathJax_Main;">,</span><span class="mspace" id="MathJax-Span-1235" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 1em;"></span><span class="mo" id="MathJax-Span-1236" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mn" id="MathJax-Span-1237" style="font-family: MathJax_Main;">2</span><span class="mo" id="MathJax-Span-1238" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 3.281em; overflow: hidden; vertical-align: -1.385em; width: 0px;"></span></span></nobr></span></div>we see that <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-94-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1239"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 25px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1240"><span class="mo" id="MathJax-Span-1241" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1242" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-1243" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span> becomes<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-95-Frame"><nobr><span class="math" id="MathJax-Span-1244"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 264px;"><span style="clip: rect(1.261em, 1000em, 2.701em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1245"><span class="msubsup" id="MathJax-Span-1246"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1247" style="font-family: MathJax_Math; font-style: italic;">u</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-1248" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1249" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1250" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-1251" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1252" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">b</span><span class="mo" id="MathJax-Span-1253" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-1254" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="msubsup" id="MathJax-Span-1255" style="padding-left: 0.222em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.892em;"><span style="clip: rect(1.702em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1256" style="font-family: MathJax_Math; font-style: italic;">v</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-1257" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1258" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1259" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-1260" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-1261" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">b</span><span class="mo" id="MathJax-Span-1262" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-1263" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-1264" style="font-family: MathJax_Main; padding-left: 0.278em;">1.</span><span class="mspace" id="MathJax-Span-1265" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 1em;"></span><span class="mo" id="MathJax-Span-1266" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1267" style="font-family: MathJax_Main;">3</span><span class="mo" id="MathJax-Span-1268" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.681em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div><br /><br />
Thus equation <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-96-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1269"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 25px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1270"><span class="mo" id="MathJax-Span-1271" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1272" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-1273" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span> gives an ellipse if <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-97-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1274"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 55px;"><span style="clip: rect(1.395em, 1000em, 2.7em, -0.341em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1275"><span class="texatom" id="MathJax-Span-1276"><span class="mrow" id="MathJax-Span-1277"><span class="mo" id="MathJax-Span-1278" style="font-family: MathJax_Main;">|</span></span></span><span class="mi" id="MathJax-Span-1279" style="font-family: MathJax_Math; font-style: italic;">b</span><span class="texatom" id="MathJax-Span-1280"><span class="mrow" id="MathJax-Span-1281"><span class="mo" id="MathJax-Span-1282" style="font-family: MathJax_Main;">|</span></span></span><span class="mo" id="MathJax-Span-1283" style="font-family: MathJax_Main; padding-left: 0.278em;"><</span><span class="mn" id="MathJax-Span-1284" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.498em; overflow: hidden; vertical-align: -0.408em; width: 0px;"></span></span></nobr></span>, a hyperbola if <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-98-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1285"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 55px;"><span style="clip: rect(1.395em, 1000em, 2.7em, -0.341em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1286"><span class="texatom" id="MathJax-Span-1287"><span class="mrow" id="MathJax-Span-1288"><span class="mo" id="MathJax-Span-1289" style="font-family: MathJax_Main;">|</span></span></span><span class="mi" id="MathJax-Span-1290" style="font-family: MathJax_Math; font-style: italic;">b</span><span class="texatom" id="MathJax-Span-1291"><span class="mrow" id="MathJax-Span-1292"><span class="mo" id="MathJax-Span-1293" style="font-family: MathJax_Main;">|</span></span></span><span class="mo" id="MathJax-Span-1294" style="font-family: MathJax_Main; padding-left: 0.278em;">></span><span class="mn" id="MathJax-Span-1295" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.498em; overflow: hidden; vertical-align: -0.408em; width: 0px;"></span></span></nobr></span>, and it reduces to a pair of lines if <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-99-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1296"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 55px;"><span style="clip: rect(1.395em, 1000em, 2.7em, -0.341em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1297"><span class="texatom" id="MathJax-Span-1298"><span class="mrow" id="MathJax-Span-1299"><span class="mo" id="MathJax-Span-1300" style="font-family: MathJax_Main;">|</span></span></span><span class="mi" id="MathJax-Span-1301" style="font-family: MathJax_Math; font-style: italic;">b</span><span class="texatom" id="MathJax-Span-1302"><span class="mrow" id="MathJax-Span-1303"><span class="mo" id="MathJax-Span-1304" style="font-family: MathJax_Main;">|</span></span></span><span class="mo" id="MathJax-Span-1305" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-1306" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.498em; overflow: hidden; vertical-align: -0.408em; width: 0px;"></span></span></nobr></span>.<br />
The question, however, is (as at the start of this article) "How could I<br />
have seen this?" We are going to change the variables <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-100-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1307"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 30px;"><span style="clip: rect(1.703em, 1000em, 2.656em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1308"><span class="mi" id="MathJax-Span-1309" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-1310" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-1311" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.019em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span> to new variables <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-101-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1312"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 29px;"><span style="clip: rect(1.702em, 1000em, 2.645em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1313"><span class="mi" id="MathJax-Span-1314" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mo" id="MathJax-Span-1315" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-1316" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">v</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.005em; overflow: hidden; vertical-align: -0.333em; width: 0px;"></span></span></nobr></span> by rotating the plane by an angle <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-102-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1317"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 9px;"><span style="clip: rect(1.44em, 1000em, 2.461em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1318"><span class="mi" id="MathJax-Span-1319" style="font-family: MathJax_Math; font-style: italic;">θ</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.111em; overflow: hidden; vertical-align: -0.083em; width: 0px;"></span></span></nobr></span>. As we do not yet know which value <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-103-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1320"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 9px;"><span style="clip: rect(1.44em, 1000em, 2.461em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1321"><span class="mi" id="MathJax-Span-1322" style="font-family: MathJax_Math; font-style: italic;">θ</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.111em; overflow: hidden; vertical-align: -0.083em; width: 0px;"></span></span></nobr></span> to take, we work with a<br />
general <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-104-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1323"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 9px;"><span style="clip: rect(1.44em, 1000em, 2.461em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1324"><span class="mi" id="MathJax-Span-1325" style="font-family: MathJax_Math; font-style: italic;">θ</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.111em; overflow: hidden; vertical-align: -0.083em; width: 0px;"></span></span></nobr></span> and make this choice later. A rotation of the plane by an angle <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-105-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1326"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 9px;"><span style="clip: rect(1.44em, 1000em, 2.461em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1327"><span class="mi" id="MathJax-Span-1328" style="font-family: MathJax_Math; font-style: italic;">θ</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.111em; overflow: hidden; vertical-align: -0.083em; width: 0px;"></span></span></nobr></span> is given by<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-106-Frame"><nobr><span class="math" id="MathJax-Span-1329"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 408px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1330"><span class="mi" id="MathJax-Span-1331" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-1332" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-1333" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">u</span><span class="mi" id="MathJax-Span-1334" style="font-family: MathJax_Main; padding-left: 0.167em;">cos</span><span class="mo" id="MathJax-Span-1335"></span><span class="mi" id="MathJax-Span-1336" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">θ</span><span class="mo" id="MathJax-Span-1337" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-1338" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">v</span><span class="mi" id="MathJax-Span-1339" style="font-family: MathJax_Main; padding-left: 0.167em;">sin</span><span class="mo" id="MathJax-Span-1340"></span><span class="mi" id="MathJax-Span-1341" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">θ</span><span class="mo" id="MathJax-Span-1342" style="font-family: MathJax_Main;">,</span><span class="mspace" id="MathJax-Span-1343" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 1em;"></span><span class="mi" id="MathJax-Span-1344" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-1345" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-1346" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">u</span><span class="mi" id="MathJax-Span-1347" style="font-family: MathJax_Main; padding-left: 0.167em;">sin</span><span class="mo" id="MathJax-Span-1348"></span><span class="mi" id="MathJax-Span-1349" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">θ</span><span class="mo" id="MathJax-Span-1350" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1351" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">v</span><span class="mi" id="MathJax-Span-1352" style="font-family: MathJax_Main; padding-left: 0.167em;">cos</span><span class="mo" id="MathJax-Span-1353"></span><span class="mi" id="MathJax-Span-1354" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">θ</span><span class="mo" id="MathJax-Span-1355" style="font-family: MathJax_Main;">.</span><span class="mspace" id="MathJax-Span-1356" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 1em;"></span><span class="mo" id="MathJax-Span-1357" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mn" id="MathJax-Span-1358" style="font-family: MathJax_Main;">4</span><span class="mo" id="MathJax-Span-1359" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div><br /><br />
If we substitute these in equation <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-107-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1360"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 25px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1361"><span class="mo" id="MathJax-Span-1362" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1363" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-1364" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span> we obtain<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-108-Frame"><nobr><span class="math" id="MathJax-Span-1365"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 448px;"><span style="clip: rect(1.261em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1366"><span class="mo" id="MathJax-Span-1367" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1368" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-1369" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1370" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">b</span><span class="mi" id="MathJax-Span-1371" style="font-family: MathJax_Main; padding-left: 0.167em;">sin</span><span class="mo" id="MathJax-Span-1372"></span><span class="mn" id="MathJax-Span-1373" style="font-family: MathJax_Main; padding-left: 0.167em;">2</span><span class="mi" id="MathJax-Span-1374" style="font-family: MathJax_Math; font-style: italic;">θ</span><span class="mo" id="MathJax-Span-1375" style="font-family: MathJax_Main;">)</span><span class="msubsup" id="MathJax-Span-1376"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1377" style="font-family: MathJax_Math; font-style: italic;">u</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-1378" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1379" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-1380" style="font-family: MathJax_Main; padding-left: 0.222em;">(</span><span class="mn" id="MathJax-Span-1381" style="font-family: MathJax_Main;">2</span><span class="mi" id="MathJax-Span-1382" style="font-family: MathJax_Math; font-style: italic;">b</span><span class="mi" id="MathJax-Span-1383" style="font-family: MathJax_Main; padding-left: 0.167em;">cos</span><span class="mo" id="MathJax-Span-1384"></span><span class="mn" id="MathJax-Span-1385" style="font-family: MathJax_Main; padding-left: 0.167em;">2</span><span class="mi" id="MathJax-Span-1386" style="font-family: MathJax_Math; font-style: italic;">θ</span><span class="mo" id="MathJax-Span-1387" style="font-family: MathJax_Main;">)</span><span class="mi" id="MathJax-Span-1388" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mi" id="MathJax-Span-1389" style="font-family: MathJax_Math; font-style: italic;">v</span><span class="mo" id="MathJax-Span-1390" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mo" id="MathJax-Span-1391" style="font-family: MathJax_Main; padding-left: 0.222em;">(</span><span class="mn" id="MathJax-Span-1392" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-1393" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mi" id="MathJax-Span-1394" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">b</span><span class="mi" id="MathJax-Span-1395" style="font-family: MathJax_Main; padding-left: 0.167em;">sin</span><span class="mo" id="MathJax-Span-1396"></span><span class="mn" id="MathJax-Span-1397" style="font-family: MathJax_Main; padding-left: 0.167em;">2</span><span class="mi" id="MathJax-Span-1398" style="font-family: MathJax_Math; font-style: italic;">θ</span><span class="mo" id="MathJax-Span-1399" style="font-family: MathJax_Main;">)</span><span class="msubsup" id="MathJax-Span-1400"><span style="display: inline-block; height: 0px; position: relative; width: 0.892em;"><span style="clip: rect(1.702em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1401" style="font-family: MathJax_Math; font-style: italic;">v</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.46em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-1402" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1403" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-1404" style="font-family: MathJax_Main; padding-left: 0.278em;">1</span><span class="mo" id="MathJax-Span-1405" style="font-family: MathJax_Main;">,</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.681em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div><br /><br />
and so if we now choose <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-109-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1406"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 9px;"><span style="clip: rect(1.44em, 1000em, 2.461em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1407"><span class="mi" id="MathJax-Span-1408" style="font-family: MathJax_Math; font-style: italic;">θ</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.111em; overflow: hidden; vertical-align: -0.083em; width: 0px;"></span></span></nobr></span> so that <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-110-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1409"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 85px;"><span style="clip: rect(1.44em, 1000em, 2.473em, -0.426em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1410"><span class="mi" id="MathJax-Span-1411" style="font-family: MathJax_Main;">cos</span><span class="mo" id="MathJax-Span-1412"></span><span class="mn" id="MathJax-Span-1413" style="font-family: MathJax_Main; padding-left: 0.167em;">2</span><span class="mi" id="MathJax-Span-1414" style="font-family: MathJax_Math; font-style: italic;">θ</span><span class="mo" id="MathJax-Span-1415" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-1416" style="font-family: MathJax_Main; padding-left: 0.278em;">0</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.128em; overflow: hidden; vertical-align: -0.099em; width: 0px;"></span></span></nobr></span>, we see that the <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-111-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1417"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 21px;"><span style="clip: rect(1.702em, 1000em, 2.462em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1418"><span class="mi" id="MathJax-Span-1419" style="font-family: MathJax_Math; font-style: italic;">u</span><span class="mi" id="MathJax-Span-1420" style="font-family: MathJax_Math; font-style: italic;">v</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.756em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> term will vanish. Thus we take <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-112-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1421"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 67px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1422"><span class="mi" id="MathJax-Span-1423" style="font-family: MathJax_Math; font-style: italic;">θ</span><span class="mo" id="MathJax-Span-1424" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mi" id="MathJax-Span-1425" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">π<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="texatom" id="MathJax-Span-1426"><span class="mrow" id="MathJax-Span-1427"><span class="mo" id="MathJax-Span-1428" style="font-family: MathJax_Main;">/</span></span></span><span class="mn" id="MathJax-Span-1429" style="font-family: MathJax_Main;">4</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>, and this with <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-113-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1430"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 25px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1431"><span class="mo" id="MathJax-Span-1432" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1433" style="font-family: MathJax_Main;">4</span><span class="mo" id="MathJax-Span-1434" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span> gives the values of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-114-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1435"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1436"><span class="mi" id="MathJax-Span-1437" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-115-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1438"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.703em, 1000em, 2.656em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1439"><span class="mi" id="MathJax-Span-1440" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.019em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span> as in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-116-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1441"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 25px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1442"><span class="mo" id="MathJax-Span-1443" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1444" style="font-family: MathJax_Main;">2</span><span class="mo" id="MathJax-Span-1445" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span> and hence the equation of the conic as in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-117-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1446"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 25px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1447"><span class="mo" id="MathJax-Span-1448" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1449" style="font-family: MathJax_Main;">3</span><span class="mo" id="MathJax-Span-1450" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span>.<br /><br />
<br /><br />
More generally, if we have an equation<br /><br />
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" style="text-align: center;"><span class="MathJax" id="MathJax-Element-118-Frame"><nobr><span class="math" id="MathJax-Span-1451"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 351px;"><span style="clip: rect(1.261em, 1000em, 2.701em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1452"><span class="mi" id="MathJax-Span-1453" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="msubsup" id="MathJax-Span-1454"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1455" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-1456" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1457" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1458" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">b</span><span class="mi" id="MathJax-Span-1459" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mi" id="MathJax-Span-1460" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-1461" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1462" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">c</span><span class="msubsup" id="MathJax-Span-1463"><span style="display: inline-block; height: 0px; position: relative; width: 0.995em;"><span style="clip: rect(1.703em, 1000em, 2.656em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1464" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.562em; position: absolute; top: -2.66em;"><span class="mn" id="MathJax-Span-1465" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1466" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1467" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">d<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-1468" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-1469" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1470" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">e</span><span class="mi" id="MathJax-Span-1471" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-1472" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1473" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">f<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-1474" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" id="MathJax-Span-1475" style="font-family: MathJax_Main; padding-left: 0.278em;">0</span><span class="mo" id="MathJax-Span-1476" style="font-family: MathJax_Main;">,</span><span class="mspace" id="MathJax-Span-1477" style="display: inline-block; height: 0em; overflow: hidden; vertical-align: 0em; width: 1em;"></span><span class="mo" id="MathJax-Span-1478" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mn" id="MathJax-Span-1479" style="font-family: MathJax_Main;">5</span><span class="mo" id="MathJax-Span-1480" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.681em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span></div>where <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-119-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1481"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 101px;"><span style="clip: rect(1.44em, 1000em, 2.656em, -0.427em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1482"><span class="mi" id="MathJax-Span-1483" style="font-family: MathJax_Math; font-style: italic;">a</span><span class="mo" id="MathJax-Span-1484" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-1485" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">b</span><span class="mo" id="MathJax-Span-1486" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-1487" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">c</span><span class="mo" id="MathJax-Span-1488" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-1489" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">d<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1490" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-1491" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">e</span><span class="mo" id="MathJax-Span-1492" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-1493" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.167em;">f<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.06em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.376em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span> are real numbers, we can try to remove the linear terms by a translation, say <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-120-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1494"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 89px;"><span style="clip: rect(1.519em, 1000em, 2.617em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1495"><span class="mi" id="MathJax-Span-1496" style="font-family: MathJax_Math; font-style: italic;">x</span><span class="mo" id="MathJax-Span-1497" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="msubsup" id="MathJax-Span-1498" style="padding-left: 0.278em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.045em;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1499" style="font-family: MathJax_Math; font-style: italic;">x</span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.613em; position: absolute; top: -2.097em;"><span class="mn" id="MathJax-Span-1500" style="font-family: MathJax_Main; font-size: 70.7%;">0</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1501" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1502" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">t</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.215em; overflow: hidden; vertical-align: -0.295em; width: 0px;"></span></span></nobr></span> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-121-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1503"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 88px;"><span style="clip: rect(1.562em, 1000em, 2.707em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1504"><span class="mi" id="MathJax-Span-1505" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span class="mo" id="MathJax-Span-1506" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="msubsup" id="MathJax-Span-1507" style="padding-left: 0.278em;"><span style="display: inline-block; height: 0px; position: relative; width: 0.937em;"><span style="clip: rect(1.703em, 1000em, 2.656em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mi" id="MathJax-Span-1508" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span><span style="left: 0.505em; position: absolute; top: -2.006em;"><span class="mn" id="MathJax-Span-1509" style="font-family: MathJax_Main; font-size: 70.7%;">0</span><span style="display: inline-block; height: 2.247em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-1510" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mi" id="MathJax-Span-1511" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;">s</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.28em; overflow: hidden; vertical-align: -0.417em; width: 0px;"></span></span></nobr></span>,<br />
and then apply the method given above. In this way, by a combination of<br />
a translation and a rotation, we can change the variables so that the <br />
conic <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-122-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1512"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 25px;"><span style="clip: rect(1.395em, 1000em, 2.701em, -0.366em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1513"><span class="mo" id="MathJax-Span-1514" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-1515" style="font-family: MathJax_Main;">5</span><span class="mo" id="MathJax-Span-1516" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.499em; overflow: hidden; vertical-align: -0.409em; width: 0px;"></span></span></nobr></span> is given in a simpler form centred at the origin with the <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-123-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1517"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 12px;"><span style="clip: rect(1.703em, 1000em, 2.462em, -0.425em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1518"><span class="mi" id="MathJax-Span-1519" style="font-family: MathJax_Math; font-style: italic;">x</span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.755em; overflow: hidden; vertical-align: -0.084em; width: 0px;"></span></span></nobr></span> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-124-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1520"><span style="display: inline-block; font-size: 136%; height: 0px; position: relative; width: 10px;"><span style="clip: rect(1.703em, 1000em, 2.656em, -0.439em); left: 0em; position: absolute; top: -2.298em;"><span class="mrow" id="MathJax-Span-1521"><span class="mi" id="MathJax-Span-1522" style="font-family: MathJax_Math; font-style: italic;">y<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.006em;"></span></span></span><span style="display: inline-block; height: 2.298em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.019em; overflow: hidden; vertical-align: -0.348em; width: 0px;"></span></span></nobr></span> axes<br />
as the axes of symmetry of the conic.<br /> <br />
<br /><br /><br />
<br /><br />
#αντικατάσταση, #Βλυκείου , #Γλυκείου , #πολυωνυμικές εξισώσεις , #ολοκληρώματα , #μετασχηματισμοί στο επίπεδοsotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-40363500825892336952017-06-03T06:12:00.001-07:002017-06-03T06:12:04.950-07:00Αλλαγή μεταβλητής 1 - Substitution method - Free math help<a href="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method.php">Substitution method - Free math help</a><br /><br />
<br /><br />
<div class="naslov"><div class="lessonIntro">Solving System of Linear Equations: (lesson 1 of 5) </div><h2>Substitution Method</h2></div><div class="prvipar">The <b>substitution method</b> is most useful for systems of 2 equations in 2 unknowns.<br />
The main idea here is that we solve one of the equations for one of the<br />
unknowns, and then substitute the result into the other equation. </div><b>Substitution method can be applied in four steps</b><br /><br />
<b>Step 1:</b><br /><br />
<div class="uvucen">Solve one of the equations for either <b>x = </b> or <b>y =</b> .</div><b>Step 2:</b><br /><br />
<div class="uvucen">Substitute the solution from step 1 into the other equation.</div><b>Step 3:</b><br /><br />
<div class="uvucen">Solve this new equation.</div><b>Step 4:</b><br /><br />
<div class="uvucen">Solve for the second variable. </div><br /><br />
<div class="e">Example 1: Solve the following system by substitution </div><div class="uvucen"><img alt="Substitution Method Example" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/1.gif" /> </div><div class="e">Solution:</div><b>Step 1:</b> Solve one of the equations for either <b>x = </b> or <b>y =</b> . We will solve <br />
second equation for y.<br /><br />
<div class="uvucen"><img alt="solution step 1" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/2.gif" /> </div><b>Step 2:</b> Substitute the solution from step 1 into the second equation.<br /><br />
<div class="uvucen"><img alt="solution step 2" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/3.gif" /> </div><b>Step 3:</b> Solve this new equation.<br /><br />
<div class="uvucen"><img alt="solution step 3" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/4.gif" /> </div><b>Step 4:</b> Solve for the second variable<br /><br />
<div class="uvucen"><img alt="solution step 4" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/5.gif" /> </div><b>The solution is: <span class="mathB">(x, y) = (10, -5)</span></b><br /><br />
<div class="e">Note: <span class="normalF">It does not matter which equation we choose<br />
first and which second. Just choose the most convenient one first! </span></div><br /><br />
<div class="googleOglasAlternativa"></div><div class="e">Example 2: <span class="blackF">Solve by substitution </span></div><div class="uvucen"><img alt="Substitution Method Example" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/6.gif" /> </div><div class="e">Solution:</div><b>Step 1: Solve one of the equations for either x = or y =.</b> <br />
Since the coefficient of y in equation 2 is -1, it is easiest to solve for y in equation 2.<br /><br />
<div class="uvucen"><img alt="solution step 1" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/7.gif" /> </div><b>Step 2:</b> Substitute the solution from step 1 into the second equation.<br /><br />
<div class="uvucen"><img alt="solution step 2" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/8.gif" /> </div><b>Step 3:</b> Solve this new equation ( for x ).<br /><br />
<div class="uvucen"><img alt="solution step 3" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/9.gif" /> </div><b>Step 4:</b> Solve for the second variable<br /><br />
<div class="uvucen"><img alt="solution step 4" src="http://www.mathportal.org/algebra/solving-system-of-linear-equations/substitution-method_files/10.gif" /> </div><b>The solution is: <span class="mathB">(x, y) = (1, 2)</span></b><br /><br />
#αντικατάσταση , #γυμνάσιο , #συστήματα sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-66787826650315542422016-11-27T10:14:00.001-08:002016-11-27T10:14:07.767-08:00Εικοσιδωδεκάεδρο Τέυχος 14 - Οκτώβρης 2016 - Πολυώνυμα<a href="http://www.mathematica.gr/forum/download/file.php?id=8868">download/file.php?id=8868</a>sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-44082601684511185882016-11-27T06:38:00.001-08:002016-11-27T06:38:52.906-08:00NON-COMMUTATIVE GR OBNER BASES FOR COMMUTATIVE ALGEBRAS<a href="http://www.math.cornell.edu/~irena/papers/grbasis.pdf">grbasis.pdf</a>sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-76005594214064448062016-10-30T08:29:00.001-07:002016-10-30T08:29:36.118-07:00THE COTES-NEWTON FACTORIZATION OF x^n +- 1<a href="http://math.nist.gov/~DLozier/SF21/SF21slides/Roy.pdf">Roy.pdf</a>sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-15726425906340838902016-10-30T08:28:00.001-07:002016-10-30T08:28:43.077-07:00Heaviside's cover up method - Διάσπαση σε απλά κλάσματα<a href="http://math.mit.edu/~jorloff/suppnotes/suppnotes03/h.pdf">h.pdf</a>sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-65816655934605544272016-06-13T14:26:00.001-07:002016-06-13T14:26:48.513-07:00A tale of two curricula: Euler's algebra text book | plus.maths.org<a href="https://plus.maths.org/content/tale-two-curricula-eulers-algebra-text-book">A tale of two curricula: Euler's algebra text book | plus.maths.org</a><br /><br />
<br /><br />
Σχολιάζοντας το «Στοιχεία Άλγεβρας» του L.Euler. sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0tag:blogger.com,1999:blog-7422685827275358894.post-46799249348793344052016-06-13T04:09:00.000-07:002016-06-13T14:27:13.865-07:00AlgebraGr - άλγεβρα<div dir="ltr" style="text-align: left;" trbidi="on">
Ό,τι αφορά την άλγεβρα!</div>
sotirishttp://www.blogger.com/profile/02913491613154020052noreply@blogger.com0