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377 lines
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377 lines
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HTML
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Factoring Quadratics</h1><br>
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<div class="center"><img src="images/quadratic-equation.svg" alt="Quadratic Equation" height="33" width="301"><br>
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<a href="quadratic-equation.html">A Quadratic Equation</a> in Standard Form<br>
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(<b>a</b>, <b>b</b>, and <b>c</b> can have any value, except that <b>a</b> can't be 0.) </div>
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<p> </p>
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<p>"Factoring" (or "Factorising" in the UK) a Quadratic is:</p>
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<p class="center larger">finding what to multiply to get the Quadratic</p>
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<div class="words">
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<p>It is called "Factoring" because we find the factors (a factor is something we multiply by)</p>
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</div>
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<div class="example">
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<h3>Example: <b>(x+4)</b> and <b>(x−1)</b> are factors of <b>x<sup>2</sup> + 3x − 4</b> </h3>
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<p class="center"><img src="images/expand-vs-factor-quadratic.svg" alt="expand vs factor quadratic" height="95" width="282"></p>
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<p>Let us "expand"<b> (x+4)</b> and <b>(x−1) </b>to be sure:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="large">(x+4)(x−1) </span></span><span class="right"><span class="large"> = x(x−1) + 4(x−1)</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= x<sup>2</sup> − x + 4x − 4</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= x<sup>2</sup> + 3x − 4 <img src="../images/style/yes.svg" alt="yes" height="30" width="30"></span></span></div>
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</div>
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<p>Yes, <b>(x+4)</b> and <b>(x−1)</b> are definitely factors of <b>x<sup>2</sup> + 3x − 4</b></p>
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</div>
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<p>Did you see that Expanding and Factoring are opposites?</p>
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<p class="center"><img src="images/expand-vs-factor-quadratic.svg" alt="expand vs factor quadratic" height="95" width="282"></p>
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<p>Expanding is usually easy, but Factoring can often be <b>tricky</b>.</p>
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<p class="center"><img src="images/factor-cake.gif" alt="factoring cake" height="55" width="219"><br>
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It is like trying to find which ingredients<br>
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went into a cake to make it so delicious.<br>
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<span class="large">It can be hard to figure out! </span></p>
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<p>OK, let's try an example where we <b>don't know</b> the factors yet:</p>
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<h2>Common Factor</h2>
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<p>First we can check for any <b>common factors</b>.</p>
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<div class="example">
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<h3>Example: what are the factors of <span class="large">6x<sup>2</sup> − 2x = 0</span> ?</h3>
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<p><b>6</b> and <b>2</b> have a common factor of <b>2</b>:</p>
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<p class="center large">2(3x<sup>2</sup> − x) = 0</p>
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<p>And <b>x<sup>2</sup></b> and <b>x</b> have a common factor of <b>x</b>:</p>
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<p class="center large">2x(3x − 1) = 0</p>
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<p>And we have done it! The factors are <b>2x</b> and <b>3x − 1</b>,</p>
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<p> </p>
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<p>We can now also find the <b>roots</b> (where it equals zero):</p>
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<ul>
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<li>2x is 0 when <b>x = 0</b></li>
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<li>3x − 1 is zero when <b>x = <span class="intbl"><em>1</em><strong>3</strong></span></b></li>
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</ul>
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<p>And this is the graph (see how it is zero at x=0 and x=<span class="intbl"><em>1</em><strong>3</strong></span>):</p>
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<p class="center"><img src="images/6x2-2x.gif" alt="graph of 6x^2 - 2x" height="200" width="252"></p>
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</div>
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<p>But it is not always that easy ...</p>
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<h2>Guess and Check</h2>
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<div class="example">
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<h3>Example: what are the factors of <span class="large">2x<sup>2</sup> + 7x + 3</span> ?</h3>
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<p>No common factors.</p>
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<p>Maybe we can <b>guess </b>an answer? Then check if we are right ... we may get lucky!</p>
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<p> </p>
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<p>Let's guess (2x+3)(x+1):</p>
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<p class="center"><span class="large">(2x+3)(x+1) = 2x<sup>2</sup> + 2x + 3x + 3<br>
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= 2x<sup>2</sup> + 5x + 3 </span>(Close but <b>WRONG</b>)</p>
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<p>How about (2x+7)(x−1):</p>
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<p class="center"><span class="large">(2x+7)(x−1) = 2x<sup>2</sup> − 2x + 7x − 7<br>
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= 2x<sup>2</sup> + 5x − 7 </span><span class="large"> </span><b>(WRONG AGAIN)</b></p>
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<p>OK, how about (2x+9)(x−1):</p>
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<p class="center"><span class="large">(2x+9)(x−1) = 2x<sup>2</sup> − 2x + 9x − 9<br>
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= 2x<sup>2</sup> + 7x − 9 </span><span class="large"> </span><b>(WRONG AGAIN!)</b></p>
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<p>We could be guessing for a <b>long time</b> before we get lucky.</p>
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</div>
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<p>That is not a very good method. So let us try something else.</p>
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<h2>A Method For Simple Cases</h2>
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<p>There is a method for simple cases.</p>
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<p>With the quadratic equation in this form:</p>
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<p class="center"><img src="images/quadratic-equation.svg" alt="Quadratic Equation" height="33" width="301"></p>
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<p><b>Step 1</b>: Find two numbers that multiply to give <span class="large">ac</span> (in other words a times c), and add to give <span class="large">b</span>.</p>
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<div class="example">
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<p>Example: <span class="large">2x<sup>2</sup> + 7x + 3</span></p>
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<p>ac is 2×3 = <b>6</b> and b is <b>7</b></p>
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<p>So we want two numbers that multiply together to make 6, and add up to 7</p>
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<p>In fact <b>6</b> and <b>1</b> do that (6×1=6, and 6+1=7)</p>
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</div>
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<div class="center80">
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<p>How do we find 6 and 1?</p>
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<p>It helps to list <a href="../numbers/factors-all-tool.html"> the factors</a> of ac=<b>6</b>, and then try adding some to get b=<b>7</b>.</p>
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<p>Factors of 6 include 1, 2, 3 and 6.</p>
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<p>Aha! 1 and 6 add to 7, and 6×1=6.</p>
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</div>
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<p><b>Step 2</b>: Rewrite the middle with those numbers:</p>
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<div class="example">
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<p>Rewrite 7x with <b>6</b>x and <b>1</b>x:</p>
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<p class="center large">2x<sup>2</sup> + <b>6x + x</b> + 3</p>
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</div>
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<p><b>Step 3</b>: Factor the first two and last two terms separately:</p>
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<div class="example">
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<p>The first two terms <span class="large">2x<sup>2</sup> + 6x</span> factor into <span class="large">2x(x+3)</span></p>
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<p>The last two terms <span class="large">x+3</span> don't actually change in this case</p>
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<p>So we get:</p>
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<p class="center large">2x(x+3) + (x+3)</p>
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</div>
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<p><b>Step 4</b>: If we've done this correctly, our two new terms should have a clearly visible common factor.</p>
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<div class="example">
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<p>In this case we can see that <span class="large">(x+3)</span> is common to both terms, so we can go:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">2x(x+3) + (x+3)</span></div>
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<div class="row"><span class="left">Which is:</span><span class="right">2x(x+3) + 1(x+3) </span></div>
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<div class="row"><span class="left">And so:</span><span class="right"><b>(2x+1)(x+3)</b></span></div>
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</div> <p>Done!</p>
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<p>Check: (2x+1)(x+3) = 2x<sup>2</sup> + 6x + x + 3 = <b>2x<sup>2</sup> + 7x + 3</b> (Yes)</p>
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</div>
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<p> </p>
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<p><b>Let's see Steps 1 to 4 again, in one go</b>:</p>
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<div class="example">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td class="large" align="center"><span><b>2x<sup>2</sup> + 7x + 3</b></span></td>
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</tr>
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<tr>
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<td class="large" align="center"><span>2x<sup>2</sup> + </span><span class="hilite">6x + x</span><span class="large"> + 3</span></td>
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</tr>
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<tr>
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<td class="large" align="center"><span class="hilite">2x(x+3)</span> + <span class="hilite">(x+3)</span></td>
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</tr>
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<tr>
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<td class="large" align="center"><span class="hilite">2x</span><span>(x+3) + </span><span class="hilite">1</span><span class="large">(x+3) </span></td>
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</tr>
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<tr>
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<td class="large" align="center"><span><b>(2x+1)(x+3)</b></span></td>
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</tr>
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</tbody></table>
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</div>
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<h3>OK, let us try another example:</h3>
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<div class="example">
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<h3>Example: <span class="large">6x<sup>2</sup> + 5x − 6</span></h3>
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<p><b>Step 1</b>: ac is 6×(−6) = <b>−36</b>, and b is <b>5</b></p>
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<p>List the positive <a href="../numbers/factors-all-tool.html">factors</a> of ac = <b>−36</b>: 1, 2, 3, 4, 6, 9, 12, 18, 36</p>
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<p>One of the numbers has to be negative to make −36, so by playing with a few different numbers I find that −4 and 9 work nicely:</p>
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<p class="center large">−4×9 = −36 and −4+9 = 5</p>
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<p> </p>
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<p><b>Step 2</b>: Rewrite <b>5x</b> with −4x and 9x:</p>
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<p class="center"><span class="large">6x<sup>2</sup> − 4x + 9x − 6</span></p>
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<p><b>Step 3</b>: Factor first two and last two:</p>
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<p class="center"><span class="large">2x(3x − 2) + 3(3x − 2)</span></p>
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<p><b>Step 4</b>: Common Factor is (3x − 2):</p>
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<p class="center larger">(2x+3)(3x − 2)</p>
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<p> </p>
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<p>Check: (2x+3)(3x − 2) = 6x<sup>2</sup> − 4x + 9x − 6 = <b>6x<sup>2</sup> + 5x − 6</b> (Yes)</p>
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</div>
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<p> </p>
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<h3>Finding Those Numbers
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</h3>
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<p>The hardest part is finding two numbers that multiply to give <span class="large">ac</span>, and add to give <span class="large">b</span>.</p>
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<p>It is partly guesswork, and it helps to <b><a href="../numbers/factors-all-tool.html">list out all the factors</a></b>.</p>
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<p>Here is another example to help you:</p>
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<div class="example">
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<h3>Example: ac = −120 and b = 7</h3>
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<p>What two numbers <b>multiply to −120</b> and <b>add to 7</b> ?</p>
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<p>The factors of 120 are (plus and minus):</p>
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<p class="center">1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120</p>
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<p>We can try pairs of factors (start near the middle!) and see if they add to 7:</p>
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<ul>
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<li>−10 x 12 = −120, and −10+12 = 2 (no)</li>
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<li>−8 x 15 = −120 and −8+15 = 7 (YES!)</li>
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</ul>
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</div>
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<h2>Get Some Practice</h2>
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<div class="center80">
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<p class="center">You can <a href="quadratic-factoring-practice.html">practice simple quadratic factoring</a>.</p>
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</div><h2>Why Factor?</h2>
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<p>Well, one of the big benefits of factoring is that we can find the <b>roots</b> of the quadratic equation (where the equation is zero).</p>
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<p>All we need to do (after factoring) is find where each of the two factors becomes zero</p>
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<div class="example">
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<h3>Example: what are the roots (zeros) of<span class="large"> 6x<sup>2</sup> + 5x − 6 </span>?</h3>
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<p>We already know (from above) the factors are</p>
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<p class="center"><span class="large">(2x + 3)(3x − 2)</span></p>
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<p>And we can figure out that</p>
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<p class="center">(2x + 3) is zero when <span class="large">x = −3/2</span></p>
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<p class="center">(3x − 2) is zero when <span class="large">x = 2/3</span></p>
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<p>So the roots of <span class="large"> 6x<sup>2</sup> + 5x − 6</span> are:</p>
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<p class="center"><span class="larger">−3/2</span> and <span class="larger">2/3</span></p>
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<p>Here is a plot of <span class="large">6x<sup>2</sup> + 5x − 6</span>, can you see where it equals zero?</p>
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<p class="center"><img src="images/factoring-quadratics-ex.gif" alt="factoring quadratics example" height="139" width="269"></p>
|
||
<p>We can also check it using a bit of arithmetic:</p>
|
||
<p><span class="large">At x = <span class="intbl"><em>−3</em><strong>2</strong></span></span>: 6(<span class="intbl"><em>−3</em><strong>2</strong></span>)<sup>2</sup> + 5(<span class="intbl"><em>−3</em><strong>2</strong></span>) − 6 = 6×(<span class="intbl"><em>9</em><strong>4</strong></span>) − <span class="intbl"><em>15</em><strong>2</strong></span> − 6 = <span class="intbl"><em>54</em><strong>4</strong></span> − <span class="intbl"><em>15</em><strong>2</strong></span> - 6 <b>= 0</b></p>
|
||
<p><span class="large">At x = <span class="intbl"><em>2</em><strong>3</strong></span></span>: 6(<span class="intbl"><em>2</em><strong>3</strong></span>)<sup>2</sup> + 5(<span class="intbl"><em>2</em><strong>3</strong></span>) − 6 = 6×(<span class="intbl"><em>4</em><strong>9</strong></span>) + <span class="intbl"><em>10</em><strong>3</strong></span> − 6 = <span class="intbl"><em>24</em><strong>9</strong></span> + <span class="intbl"><em>10</em><strong>3</strong></span> - 6 <b>= 0</b></p>
|
||
</div>
|
||
<h2><br></h2>
|
||
<h2>Graphing
|
||
|
||
</h2>
|
||
<p>We can also try <a href="../data/function-grapher.html">graphing the quadratic equation</a>. Seeing where it equals zero can give us clues.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: (continued)</h3>
|
||
<p>Starting with <span class="large">6x<sup>2</sup> + 5x − 6</span> and <b>just this plot:</b></p>
|
||
<p class="center"><img src="images/factoring-quadratics-ex.gif" alt="factoring quadratics example" height="139" width="269"></p>
|
||
<p>The roots are <b>around</b> x = −1.5 and x = +0.67, so we can <b>guess</b> the roots are:</p>
|
||
<p class="center"><span class="large">−3/2</span> and <span class="large">2/3</span></p>
|
||
<p>Which can help us work out the factors <b>2x + 3</b> and <b>3x − 2</b></p>
|
||
<p>Always check though! The graph value of +0.67 might not really be 2/3</p>
|
||
</div>
|
||
|
||
|
||
<h2>General Solution</h2>
|
||
<p>Quadratic equations have symmetry, the left and right are like mirror images:</p>
|
||
<p class="center">
|
||
|
||
<img src="images/quadratic-mid.svg" alt="quadratic mid line" title="quadratic mid line"> </p>
|
||
<p>The midline is at <b>−b/2</b>, and we can calculate the value <b>w</b> with these steps:</p>
|
||
<ul>
|
||
<li>First, "a" must be 1, if not then divide b and c by a: </li>
|
||
<ul>
|
||
<li>b = b/a, c = c/a</li></ul>
|
||
<li>mid = −b/2</li>
|
||
<li>w = √(mid<sup>2</sup> − c)</li>
|
||
<li>roots are at mid−w and mid+w</li></ul>
|
||
<div class="example">
|
||
|
||
<h3>Example:<b> x<sup>2</sup> + 3x − 4</b></h3>
|
||
|
||
<p class="center">
|
||
a = 1, b = 3 and c = −4
|
||
</p>
|
||
<ul>
|
||
<li>a= 1, so we can go to next step</li>
|
||
<li>mid = −<span class="intbl"><em>3</em><strong>2</strong></span></li>
|
||
<li>w = √[(<span class="intbl"><em>3</em><strong>2</strong></span>)<sup>2</sup> − (−4)] = √(<span class="intbl"><em>9</em><strong>4</strong></span> + 4) = √<span class="intbl"><em>25</em><strong>4</strong></span> = <b><span class="intbl"><em>5</em><strong>2</strong></span></b></li>
|
||
<li>roots are at −<span class="intbl"><em>3</em><strong>2</strong></span>−<span class="intbl"><em>5</em><strong>2</strong></span> = <b>−4</b> and −<span class="intbl"><em>3</em><strong>2</strong></span>+<span class="intbl"><em>5</em><strong>2</strong></span> = <b>1</b> </li></ul>
|
||
<p>So we can factor <b>x<sup>2</sup> + 3x − 4</b> into <b>(x + 4)(x </b><b>− 1) </b></p>
|
||
</div>
|
||
|
||
<h2>Quadratic formula</h2>
|
||
|
||
<p>We can also use the <a href="quadratic-equation.html">quadratic formula</a>:</p>
|
||
<p class="center"><img src="images/quadratic-formula.svg" alt="Quadratic Formula" height="79" width="286"></p>
|
||
<p>We get two answers <span class="large">x<sub>+</sub></span> and <span class="large">x<sub>−</sub></span> (one is for the "+" case, and the other is for the "−" case in the "±") that gets us this factoring:</p>
|
||
<p class="center larger">a(x − x<sub>+</sub>)(x − x<sub>−</sub>)</p>
|
||
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: what are the roots of<span class="large"> 6x<sup>2</sup> + 5x − 6 </span>?</h3>
|
||
<p>Substitute a=6, b=5 and c=−6 into the formula:</p>
|
||
<p class="center large">x = <span class="intbl"><em>−b ± √(b<sup>2</sup> − 4ac)</em><strong>2a</strong></span></p>
|
||
<p class="center large">= <span class="intbl"><em>−5 ± √(5<sup>2</sup> − 4×6×(−6))</em><strong>2×6</strong></span></p>
|
||
<p class="center large">= <span class="intbl"><em>−5 ± √(25 + 144)</em><strong>12</strong></span></p>
|
||
<p class="center large">= <span class="intbl"><em>−5 ± √169</em><strong>12</strong></span></p>
|
||
<p class="center large">= <span class="intbl"><em>−5 ± 13</em><strong>12</strong></span></p>
|
||
<p>So the two roots are:</p>
|
||
<p class="center large">x<sub>+</sub> = (−5 <span class="hilite">+</span> 13) / 12 = 8/12 = 2/3,</p>
|
||
<p class="center large">x<sub>−</sub> = (−5 <span class="hilite">−</span> 13) / 12 = −18/12 = −3/2</p>
|
||
<p>(Notice that we get the same answer as when we did the factoring earlier.)</p>
|
||
<p> </p>
|
||
<p>Now put those values into <span class="large">a(x − x<sub>+</sub>)(x − x<sub>−</sub>)</span>:</p>
|
||
<p class="center large">6(x − 2/3)(x + 3/2)</p>
|
||
<p>We can rearrange that a little to simplify it:</p>
|
||
<p class="center large">3(x − 2/3) × 2(x + 3/2) = (3x − 2)(2x + 3)</p>
|
||
<p>Done!</p>
|
||
</div><br><div class="questions">362, 1203, 2262, 363, 1204, 2263, 2100, 2101, 2102, 2103, 2264, 2265</div>
|
||
|
||
<div class="related">
|
||
<a href="factoring.html">Factoring - Introduction</a>
|
||
<a href="quadratic-equation.html">Quadratic Equations</a>
|
||
<a href="completing-square.html">Completing the Square</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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