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<h4>Recognizing the Expanded Product of the Difference of Two Squares</h4>
<p>Mathematically, something special happens when two binomials that are conjugates are multiplied. In other words, the product of an expression of the form (\(a - b)(a + b)\) results in a unique expression called a difference of two squares. Here are some examples of these special products.</p>
<p><strong>Example 1</strong></p>
<p>Let's look at the expression \((12 + 5)(12 − 5)\). We want to determine if it is equivalent to \(12^2−5^2\).</p>
<p>We can use the distributive property of multiplication over addition, or FOIL process, to multiply.</p>
<p>\((12 + 5)(12 − 5)\) <br>
<br>
\(\begin{array}{rcl} &=&12^2-12\left(5\right)+5\left(12\right)-5^2\\&&\\&=&12^2-60+60-5^2\\&&\\&=&12^2-5^2\\&&\end{array}\)<br>
So, \((12 + 5)(12 − 5)\) is equivalent to \(12^2-5^2\).
</p>
<p><strong>Example 2</strong></p>
<p>Let's look at a different example using a <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">variable</span>.</p>
<p>\((x + 7)(x − 7)\)</p>
<p>Is this equivalent to \(x^2−7^2\)?</p>
<p>\((x + 7)(x − 7)\)<br>
<br>
\(\begin{array}{rcl} &=&x^2-7x+7x-7^2\\&&\\&=&x^2-7^2=x^2-49\\&&\end{array}\)<br>
This can also be shown using a diagram.
</p>
<table class="os-raise-doubleheadertable">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">
\(x\)
</th>
<th scope="col">
–7
</th>
</tr></thead><tbody>
<tr>
<th scope="row">
\(x\)
</td>
<td>
\(x^2\)
</td>
<td>
\(-7x\)
</td>
</tr>
<tr>
<th scope="row">
7
</th>
<td>
\(7x\)
</td>
<td>
\(-7(7)\)
</td>
</tr>
</tbody>
</table>
<br>
<p>You have studied this process earlier when you learned the difference of squares.</p>
<p>In this lesson, we will reverse this process to rewrite the <span class="os-raise-ib-tooltip" data-schema-version="“1.0”" data-store="glossary-tooltip">standard form</span> of a <span class="os-raise-ib-tooltip" data-schema-version="“1.0”" data-store="glossary-tooltip">quadratic expression</span> into its <span class="os-raise-ib-tooltip" data-schema-version="“1.0”" data-store="glossary-tooltip">factored form</span>.</p>
<p>It is important to remember the <span class="os-raise-ib-tooltip" data-schema-version="“1.0”" data-store="glossary-tooltip">linear term</span>, or middle term of a quadratic trinomial, equals zero only holds true for the <span class="os-raise-ib-tooltip" data-schema-version="“1.0”" data-store="glossary-tooltip">difference of squares</span>. It does not work for an expression such as \(x^2+25\). (In fact, there is no way to multiply two binomials that would result in this expression.)</p>
<p><strong>Example 3</strong></p>
<p>Is \((x + 5)^2\) equivalent to \(x^2+25\)?</p>
<p>Let's find the equivalent expression for \((x + 5)^2\) or \((x + 5)(x + 5)\).</p>
<p>\(\begin{array}{rcl}\left(x+5\right)^2&=&\left(x+5\right)\left(x+5\right)\\&&\\&=&x^2+5x+5x+25\\&&\\&=&x^2+10x+25\\&&\end{array}\)<br>
Since \(x^2+ 10x + 25\) is not equivalent to \(x^2+25\), these expressions are not the same.</p>
<h4>Try It: Recognizing the Expanded Product of the Difference of Two Squares</h4>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<li>Is the expression \((8 − 7)(8 + 7)\) equivalent to the expression \(8^2−7^2\)? Show your reasoning using the distributive property, or FOIL.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Here is how to recognize the expanded product of the difference of two squares:</p>
<p>Yes. Here is how to use the difference of squares to show equivalency between the expressions:</p>
<p>\((8 − 7)(8 + 7) = 8^2+ 56 − 56 − 7^2= 8^2 − 7^2\)</p>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li>Is \((x + 4)(x + 4)\) equivalent to \(x^2+ 4^2\)? Support your answer with a diagram.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal2">
<p>Here is how to recognize the expanded product of the difference of two squares:</p>
<p>No. The expressions are not equivalent.</p>
<table class="os-raise-doubleheadertable">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">
\(x\)
</th>
<th scope="col">
4
</th>
</tr></thead><tbody>
<tr>
<th scope="row">
\(x\)
</th>
<td>
\(x^2\)
</td>
<td>
\(4x\)
</td>
</tr>
<tr>
<th scope="row">
4
</th>
<td>
\(4x\)
</td>
<td>
<p>\(4(4)\)</p>
</td>
</tr>
</tbody>
</table>
<br>
<p>\((x + 4)(x + 4) = x^2 + 8x + 4^2 = x^2 + 8x + 16\)</p>
<p>It is not equivalent to \(x^2 + 4^2 = x^2 + 16\).</p>
</div>