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153 lines
6.1 KiB
HTML
153 lines
6.1 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/algebra/conjugate.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:36:15 GMT -->
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<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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<title>Conjugate</title>
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<meta name="description" content="The conjugate is where we change the sign in the middle of two terms">
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<div id="hdr"></div>
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<div id="cookOK"></div>
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</header>
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<div class="mid">
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<nav>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Conjugate</h1>
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<p>The conjugate is where we <b>change the sign in the middle</b> of two terms like this:</p>
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<p class="center"><img src="images/conjugate.svg" alt="conjugate of 3x+1 is 3x-1" height="138" width="227"></p>
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<div class="beach">
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We only use it in expressions
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with <b>two terms</b>, called "binomials":
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr style="text-align:center;">
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<td><img src="images/binomial.svg" alt="Binomial" height="97" width="110"></td>
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</tr>
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<tr style="text-align:center;">
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<td>example of a binomial</td>
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</tr>
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</tbody></table>
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</div>
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<p>Here are some more examples:</p>
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<table align="center" cellpadding="6" border="0">
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<tbody>
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<tr style="text-align:center;">
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<th class="large">Expression</th>
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<th class="large"> </th>
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<th class="large">Its Conjugate</th>
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</tr>
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<tr style="text-align:center;">
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<td class="large">x<sup>2</sup> − 3</td>
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<td class="large"> ⇔ </td>
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<td class="large">x<sup>2</sup> + 3</td>
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</tr>
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<tr style="text-align:center;">
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<td class="large">a + b</td>
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<td class="large"> ⇔ </td>
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<td class="large">a − b</td>
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</tr>
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<tr style="text-align:center;">
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<td class="large">a − b<sup>3</sup></td>
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<td class="large"> ⇔ </td>
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<td class="large">a + b<sup>3</sup></td>
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</tr>
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</tbody></table>
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<h2>Examples of Use</h2>
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<p>The conjugate can be very useful because ...</p>
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<p class="center">... when we multiply something by its conjugate we get <b>squares</b> like this:</p>
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<div class="center larger">(a+b)(a−b) = a<sup>2</sup> − b<sup>2</sup></div>
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<!-- (a+b)(a-b) = a^2 - b^2 -->
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<h3>How does that help?</h3>
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<p>It can help us move a square root from the bottom of a fraction (the <i>denominator</i>) to the top, or vice versa. Read <a href="rationalize-denominator.html">Rationalizing the Denominator</a> to find out more:</p>
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<div class="example">
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<h3><b>Example:</b> Move the square root of 2 to the top:<br>
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<br>
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<div class="center large"><span class="intbl"><em>1</em><strong>3−√2</strong></span></div></h3>
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<p>We can <i><b>multiply both top and bottom by 3+√2 (the conjugate of 3−√2)</b></i>, which won't change the value of the fraction:</p>
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<p class="center large"><span class="intbl"><em>1</em><strong>3−√2</strong></span> × <span class="intbl"><em>3+√2</em><strong>3+√2</strong></span> = <span class="intbl"><em>3+√2</em><strong>3<sup>2</sup>−(√2)<sup>2</sup></strong></span> = <span class="intbl"><em>3+√2</em><strong>7</strong></span></p>
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<p class="center">(The denominator becomes <b>(a+b)(a−b) = a<sup>2</sup> − b<sup>2</sup></b> which simplifies to 9−2=7)</p>
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<p> </p>
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<p><i>Use a <a href="../numbers/calculator.html">calculator</a> to work out the value before and after ... is it the same?</i></p>
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</div>
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<p> </p>
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<p>So try to remember this little trick, it may help you solve an equation one day!</p>
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<p> </p>
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<div class="questions">354, 3192, 2258, 3193, 355, 3194, 2259, 3195, 3892, 3893</div>
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<div class="related">
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<a href="rationalize-denominator.html">Rationalizing the Denominator</a>
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<a href="index.html">Algebra Index</a>
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</div>
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</article>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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