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355 lines
18 KiB
HTML
355 lines
18 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Completing the Square</h1>
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<p>"<b>Completing the Square</b>" is where we ...</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td> ... take a <a href="quadratic-equation.html">Quadratic Equation</a><br>
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like this:</td>
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<td rowspan="2"><img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></td>
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<td>and turn it<br>
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into this:</td>
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</tr>
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<tr style="text-align:center;">
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<td><span class="large" style="white-space: nowrap;">ax<sup>2</sup> + bx + c = 0</span></td>
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<td><span class="large" style="white-space: nowrap;">a(x+<i>d</i>)<sup>2</sup> + <i>e</i> = 0</span></td>
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</tr>
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</tbody></table>
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</div>
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<p><br></p>
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<div class="tbl">
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<div class="row"><span class="left">For those of you in a hurry, I can tell you that: </span><span class="right"><span class="center large">d = <span class="intbl"><em>b</em><strong>2a</strong></span></span></span></div>
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<div class="row"><span class="left">and:</span><span class="right"><span class="center large">e = c − <span class="intbl"><em>b<sup>2</sup></em><strong>4a</strong></span></span></span></div>
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</div>
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<p><br>
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But if you have time, let me show you how to "<b>Complete the Square</b>" yourself.</p>
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<h2>Completing the Square</h2>
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<p>Say we have a simple expression like <span class="large">x<sup>2</sup> + bx</span>. Having <span class="large">x</span> twice in the same expression can make life hard. What can we do?</p>
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<p>Well, with a little inspiration from Geometry we can convert it, like this:</p>
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<p class="center"><img src="images/completing-square-geometry.svg" alt="Completing the Square Geometry" style="max-width:100%" height="156" width="656"></p>
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<p>As you can see <span class="large">x<sup>2</sup> + bx</span> can be rearranged <i><b>nearly</b></i> into a square ...</p>
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<p class="center">... and we can <b>complete the square</b> with <span class="large">(b/2)<sup>2</sup> </span></p>
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<p>In Algebra it looks like this:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td><span class="larger">x<sup>2</sup> + bx</span></td>
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<td><span class="larger">+ (b/2)<sup>2</sup></span></td>
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<td style="width:50px;"><span class="large">=</span></td>
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<td><span class="larger">(x+b/2)<sup>2</sup></span></td>
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</tr>
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<tr style="text-align:center;">
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<td> </td>
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<td>"Complete the<br>
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Square"</td>
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<td style="width:50px;"> </td>
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<td> </td>
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</tr>
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</tbody></table> </div>
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<p>So, by adding <span class="large">(b/2)<sup>2</sup></span> we can complete the square.</p>
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<p class="center">The result of <span class="larger">(x+b/2)<sup>2</sup></span> has <span class="large">x</span> only <b>once</b>, which is easier to use.</p>
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<h2>Keeping the Balance</h2>
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<p>Now ... we can't just <i><b>add</b></i> <span class="large">(b/2)<sup>2</sup></span> without also <i><b>subtracting</b></i> it too! Otherwise the whole value changes.</p>
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<p>So let's see how to do it properly with an example:</p>
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<table style="margin:auto;">
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<tbody>
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<tr>
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<td>Start with: </td>
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<td><img src="images/completing-square-example1.svg" alt="x^2 + 6x + 7" height="25" width="137"></td>
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</tr>
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<tr>
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<td> </td>
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<td>("b" is 6 in this case)</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td colspan="2">Complete the Square:</td>
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</tr>
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<tr>
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<td>
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<p> </p></td>
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<td><img src="images/completing-square-insert.svg" alt="x^2 + 6x + (6/2)^2 + 7 - (6/2)^2 " height="110" width="307">
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<p align="right">Also <b>subtract</b> the new term</p></td>
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</tr>
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<tr>
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<td colspan="2">
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<p>Simplify it and we are done.</p></td>
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</tr>
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<tr>
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<td> </td>
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<td><img src="images/completing-square-simplify.svg" alt="simplifies to (x+3)^2" height="138" width="525"></td>
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</tr>
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</tbody></table>
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<p>The result:</p>
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<p class="center larger">x<sup>2</sup> + 6x + 7 = (x+3)<sup>2</sup> − 2</p>
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<p class="center">And now <span class="large">x</span> only appears once, and our job is done!</p>
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<h2>A Shortcut Approach</h2>
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<p>Here is a quick way to get an answer. You may like this method.</p>
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<p>First think about the result we want: <span class="large">(x+d)<sup>2</sup> + e</span></p>
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<p>After <a href="special-binomial-products.html">expanding</a> (x+d)<sup>2</sup> we get: <span class="large">x<sup>2</sup> + 2dx + d<sup>2</sup> + e</span></p>
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<p>Now see if we can turn our example into that form to discover d and e</p>
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<div class="example">
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<h3>Example: try to fit <span class="center large">x<sup>2</sup> + 6x + 7</span> into <span class="large">x<sup>2</sup> + 2dx + d<sup>2</sup> + e</span></h3>
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<p class="center"><img src="images/completing-square-expanded.svg" alt="x^2 + (6x) + [7] matches x^2 + (2dx) + [d^2+e]" style="max-width:100%" height="119" width="413"></p>
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<p>Now we can "force" an answer:</p>
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<ul>
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<li>We know that <span class="large">6x</span> must end up as <span class="large">2dx</span>, so <span class="large"><b>d</b></span><b> must be 3</b></li>
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<li>Next we see that <span class="large">7</span> must become d<sup>2</sup> + e = <span class="large">9 + e</span>, so <span class="large"><b>e</b></span><b> must be −2</b></li>
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</ul>
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<p>And we get the same result <span class="large">(x+3)<sup>2</sup> − 2</span> as above!</p>
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</div>
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<p> </p>
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<p>Now, let us look at a useful application: solving Quadratic Equations ...</p>
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<h2>Solving General Quadratic Equations by Completing the Square</h2>
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<p>We can complete the square to <b>solve</b> a <a href="quadratic-equation.html">Quadratic Equation</a> (find where it is equal to zero).</p>
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<p>But a general Quadratic Equation can have a <a href="definitions.html">coefficient</a> of <span class="large">a</span> in front of <span class="large">x<sup>2</sup></span>:</p>
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<p class="center large">ax<sup>2</sup> + bx + c = 0</p>
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<p>But that is easy to deal with ... just divide the whole equation by "a" first, then carry on:</p>
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<p class="center large">x<sup>2</sup> + (b/a)x + c/a = 0</p>
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<h2>Steps</h2>
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<p>Now we can <b>solve</b> a Quadratic Equation in 5 steps:</p>
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<div class="bigul">
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<ul>
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<li><b>Step 1</b> Divide all terms by <b>a</b> (the coefficient of <b>x<sup>2</sup></b>).</li>
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<li><b>Step 2</b> Move the number term (<b>c/a</b>) to the right side of the equation.</li>
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<li><b>Step 3</b> Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.</li>
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</ul>
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<p>We now have something that looks like (x + p)<sup>2</sup> = q, which can be solved rather easily:</p>
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<ul>
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<li><b>Step 4</b> Take the square root on both sides of the equation.</li>
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</ul>
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<ul>
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<li><b>Step 5</b> Subtract the number that remains on the left side of the equation to find <b>x</b>.</li>
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</ul>
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</div>
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<h2>Examples</h2>
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<p>OK, some examples will help!</p>
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<div class="example">
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<h3>Example 1: Solve x<sup>2</sup> + 4x + 1 = 0</h3>
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<p><b>Step 1</b> can be skipped in this example since the coefficient of x<sup>2</sup> is 1</p>
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<p><b>Step 2</b> Move the number term to the right side of the equation:</p>
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<div class="so">x<sup>2</sup> + 4x = -1</div>
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<p><b>Step 3</b> Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.</p>
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<p>(b/2)<sup>2</sup> = (4/2)<sup>2</sup> = 2<sup>2</sup> = 4</p>
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<div class="so">x<sup>2</sup> + 4x + 4 = -1 + 4</div>
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<div class="so">(x + 2)<sup>2</sup> = 3</div>
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<p><b>Step 4</b> Take the square root on both sides of the equation:</p>
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<div class="so"> x + 2 = ±√3 = ±1.73 (to 2 decimals)</div>
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<p><b>Step 5</b> Subtract 2 from both sides:</p>
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<div class="so"> x = ±1.73 – 2 = -3.73 or -0.27 </div>
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</div>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>
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<p>And here is an interesting and useful thing.</p>
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<p>At the end of step 3 we had the equation:</p>
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<div class="so">(x + <span class="hilite">2</span>)<sup>2</sup> = <span class="hilite">3</span></div>
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<p class="center">It gives us the <b>vertex</b> (turning point) of x<sup>2</sup> + 4x + 1: <b>(-2, -3)</b></p></td>
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<td> </td>
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<td><img src="images/completing-square-graph2.gif" alt="graph" height="160" width="161"></td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="example">
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<h3>Example 2: Solve 5x<sup>2</sup> – 4x – 2 = 0</h3>
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<p><b>Step 1</b> Divide all terms by 5</p>
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<div class="so"> x<sup>2</sup> – 0.8x – 0.4 = 0</div>
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<p><b>Step 2</b> Move the number term to the right side of the equation:</p>
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<div class="so"> x<sup>2</sup> – 0.8x = 0.4</div>
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<p><b>Step 3</b> Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:</p>
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<p>(b/2)<sup>2</sup> = (0.8/2)<sup>2</sup> = 0.4<sup>2</sup> = 0.16</p>
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<div class="so">x<sup>2</sup> – 0.8x + 0.16 = 0.4 + 0.16</div>
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<div class="so">(x – 0.4)<sup>2</sup> = 0.56</div>
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<p><b>Step 4</b> Take the square root on both sides of the equation:</p>
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<div class="so"> x – 0.4 = ±√0.56 = ±0.748 (to 3 decimals)</div>
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<p><b>Step 5</b> Subtract (-0.4) from both sides (in other words, add 0.4):</p>
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<div class="so"> x = ±0.748 + 0.4 = -0.348 or 1.148</div>
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</div>
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<h2>Why "Complete the Square"?</h2>
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<p>Why complete the square when we can just use the <a href="../quadratic-equation-solver.html">Quadratic Formula</a> to solve a Quadratic Equation?</p>
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<div class="indent50px">
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<p>Well, one reason is given above, where the new form not only shows us the vertex, but makes it easier to solve.</p>
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<p>There are also times when the form <b>ax<sup>2</sup> + bx + c</b> may be part of a <b>larger</b> question and rearranging it as <b>a(x+<i>d</i>)<sup>2</sup> + <i>e</i></b> makes the solution easier, because <b>x</b> only appears once.</p>
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<p>For example "x" may itself be a function (like <i>cos(z)</i>) and rearranging it may open up a path to a better solution.</p>
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<p>Also Completing the Square is the first step in the <a href="quadratic-equation-derivation.html">Derivation of the Quadratic Formula</a></p>
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</div>
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<p>Just think of it as another tool in your mathematics toolbox.</p>
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<p> </p>
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<div class="questions">364, 1205, 365, 2331, 2332, 3213, 3896, 3211, 3212, 1206</div>
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<p> </p>
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<div class="center80">
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<h3>Footnote: Values of "d" and "e"</h3>
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<p>How did I get the values of <b>d</b> and <b>e</b> from the top of the page?</p>
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<div class="beach">
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<table align="center" cellpadding="4" border="0">
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<tbody>
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<tr>
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<td>Start with</td>
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<td style="background-color:white;"><b><img src="images/latex/cs-h.gif" alt="equn" height="17" width="118"></b></td>
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</tr>
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<tr>
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<td>Divide the equation by <b>a</b></td>
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<td style="background-color:white;"><b><img src="images/latex/cs-i.gif" alt="equn" height="35" width="119"></b></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Put <b>c/a</b> on other side</td>
|
||
<td style="background-color:white;"><b><img src="images/latex/cs-j.gif" alt="equn" height="35" width="103"></b></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Add <b>(b/2a)<sup>2</sup></b> to both sides</td>
|
||
<td style="background-color:white;"><b><sup><img src="images/latex/cs-k.gif" alt="equn" height="46" width="240"></sup></b></td>
|
||
</tr>
|
||
<tr>
|
||
<td colspan="2"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td>"Complete the Square"</td>
|
||
<td style="background-color:white;"><b> <sup><img src="images/latex/cs-l.gif" alt="equn" height="46" width="190"></sup></b></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Now bring everything back...</td>
|
||
<td><br>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>... to the left side</td>
|
||
<td style="background-color:white;"><b><img src="images/latex/cs-m.gif" alt="equn" height="46" width="206"></b></td>
|
||
</tr>
|
||
<tr>
|
||
<td>... to the original multiple <b>a</b> of x<sup>2</sup></td>
|
||
<td style="background-color:white;"><b><img src="images/latex/cs-n.gif" alt="equn" height="46" width="187"></b></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div><br>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"> And you will notice that we have: </span><span class="right">
|
||
<div class="large">a(x+d)<sup>2</sup> + e = 0</div></span></div>
|
||
<div class="row"><span class="left">Where:</span><span class="right"><span class="center large">d = <span class="intbl">
|
||
<em>b</em>
|
||
<strong>2a</strong>
|
||
</span></span></span></div>
|
||
<div class="row"><span class="left">and:</span><span class="right"><span class="center large">e = c − <span class="intbl">
|
||
<em>b<sup>2</sup></em>
|
||
<strong>4a</strong>
|
||
</span></span></span></div>
|
||
<div class="row"><span class="left">Just like at the top of the page!</span></div>
|
||
</div>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="quadratic-equation.html">Quadratic Equations</a>
|
||
<a href="factoring-quadratics.html">Factoring Quadratics</a>
|
||
<a href="quadratic-equation-graphing.html">Graphing Quadratic Equations</a>
|
||
<a href="quadratic-equation-real-world.html">Real World Examples of Quadratic Equations</a>
|
||
<a href="quadratic-equation-derivation.html">Derivation of Quadratic Equation</a>
|
||
<a href="../quadratic-equation-solver.html">Quadratic Equation Solver</a>
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<a href="index.html">Algebra Index</a>
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