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52286e6b-445f-4f83-88df-369d69257459.html
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<h4>Using Factored Form and the Zero Product Property to Solve Quadratic Equations</h4>
<p>We have seen <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">quadratic equations</span> in many different forms. Let’s look at an example in which we find the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">factored form</span> and then use the <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">zero product property</span> to solve.</p>
<p><strong>Example 1</strong></p>
<p>Solve \(x^2 + 3x − 50 = -10\).</p>
<p><strong>Step 1</strong> - Set the equation equal to 0 by adding 10 to both sides.</p>
<p>\(x^2+ 3x − 40 = 0\)</p>
<p><strong>Step 2</strong> - Find the factored form.</p>
<p> Which numbers have a product of –40 and a sum of +3?</p>
<p>\((x + 8)(x − 5) = 0\)</p>
<p><strong>Step 3</strong> - Use the zero product property to solve.</p>
<p> Set each factor equal to 0 and solve.</p>
<p>\((x + 8) = 0\;\;\;\;(x − 5) = 0\)</p>
<p>\(x = -8\;\;\;\;\;\;\;\;x = 5\)</p>
<p>The solution is \(x = -8\) and \(x = 5\).</p>
<br>
<p><strong>Example 2</strong></p>
<p>Solve \(x^2+ 12x + 21 = -15\).</p>
<p><strong>Step 1</strong> - Set the equation equal to 0 by adding 15 to both sides.</p>
<p>\(x^2 + 12x + 36 = 0\)</p>
<p><strong>Step 2</strong> - Find the factored form.</p>
<p> Which numbers have a product of 36 and a sum of 12?</p>
<p>\((x + 6)(x + 6) = 0\)</p>
<p><strong>Step 3</strong> - Use the zero product property to solve.</p>
<p> Set each factor equal to 0 and solve. The factors are the same, so there is one solution.</p>
<p>\((x + 6) = 0\;\;\;\;(x + 6) = 0\)</p>
<p> \(x = -6\;\;\;\;\;\;\;\;x = -6\) </p>
<p>The only solution is \(x = -6\).</p>
<br>
<h4>Try It: Using Factored Form and the Zero Product Property to Solve Quadratic Equations</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>Solve \(z^2− 5z − 48 = -12\).</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 solve the quadratic equations using the factored form and the zero product property:</p>
<p>\(z^2− 5z − 48 = -12\)</p>
<p><strong>Step 1</strong> - Set the equation equal to 0.</p>
<p> \(z^2− 5z − 36 = 0\)</p>
<p><strong>Step 2</strong> - Find the factored form.</p>
<p> \((z − 9)(z + 4) = 0\) </p>
<p><strong>Step 3</strong> - Use the zero product property to solve.</p>
<p> \(z = 9\) and \(z = -4\) </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>Solve \(p^2− 3p − 108 = -20\).</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 solve the quadratic equations using the factored form and the zero product property:</p>
<p>\(p^2− 3p − 108 = -20\)</p>
<p><strong>Step 1</strong> - Set the equation equal to 0.</p>
<p> \(p^2− 3p − 88 = 0\) </p>
<p><strong>Step 2</strong> - Find the factored form.</p>
<p> \((p − 11)(p + 8) = 0\) </p>
<p><strong>Step 3</strong> - Use the zero product property to solve.</p>
<p> \(p = 11\) and \(p = -8\)</p>
</div>