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HTML
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<title>Derivation of Quadratic Formula</title>
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<h1 align="center">Derivation of Quadratic Formula</h1>
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<p>A <a href="quadratic-equation.html">Quadratic Equation</a> looks like this:</p>
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<p align="center"><img src="images/quadratic-equation.svg" alt="Quadratic Equation: ax^2 + bx + c = 0" /></p>
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<p>And it can be <a href="../quadratic-equation-solver.html">solved</a> using the Quadratic Formula:</p>
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<p align="center"><img src="images/quadratic-formula.svg" alt="Quadratic Formula: x = [ -b (+-) sqrt(b^2 - 4ac) ] / 2a" /></p>
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<p>That formula looks like magic, but you can follow the steps to see how it comes about.</p>
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<h2>1. Complete the Square</h2>
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<p><span class="large">ax<sup>2</sup> + bx + c </span>has "x" in it twice, which is hard to solve.</p>
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<p>But there is a way to rearrange it so that "x" only appears once. It is called <a href="completing-square.html">Completing the Square</a> (please read that first!).</p>
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<p>Our aim is to get something like <i><b>x<sup>2</sup> + 2dx + d<sup>2</sup></b></i>, which can then be simplified to <i><b>(x+d)<sup>2</sup></b></i></p>
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<p> So, let's go:</p>
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<div class="beach">
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<table width="90%" border="0" align="center">
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<tr>
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<td>Start with</td>
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<td><b><img src="images/latex/cs-h.gif" width="118" height="17" alt="ax^2 + bx + c=0" /></b></td>
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</tr>
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<tr>
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<td>Divide the equation by a</td>
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<td><b><img src="images/latex/cs-i.gif" width="119" height="35" alt="x^2 + bx/a + c/a = 0" /></b></td>
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</tr>
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<tr>
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<td>Put c/a on other side</td>
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<td><b><img src="images/latex/cs-j.gif" width="103" height="35" alt="x^2 + bx/a = -c/a" /></b></td>
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</tr>
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<tr>
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<td>Add (b/2a)<sup>2</sup> to both sides</td>
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<td><b><sup><img src="images/latex/cs-k.gif" width="240" height="46" alt="x^2 + bx/a + (b/2a)^2 = -c/a + (b/2a)^2" /></sup></b></td>
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</tr>
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<tr>
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<td colspan="2">
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<div align="center"> <i><br />
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The <b>left hand side</b> is now in the <b>x<sup>2</sup> + 2dx + d<sup>2</sup></b> format,
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where "d" is "b/2a"<br />
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So we can re-write it this way:<br />
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<br />
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</i></div>
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</td>
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</tr>
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<tr>
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<td>"Complete the Square"</td>
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<td><b> <sup><img src="images/latex/cs-l.gif" width="190" height="46" alt="(x+b/2a)^2 = -c/a + (b/2a)^2" /></sup></b></td>
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</tr>
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</table>
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</div>
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<p>Now x only appears once and we are making progress.</p>
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<h2>2. Now Solve For "x"</h2>
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<p>Now we just need to rearrange the equation to leave "x" on the left
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<br /> </p>
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<table width="90%" border="0" align="center" cellspacing="4">
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<tr>
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<td>Start with</td>
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<td><b> <sup><img src="images/latex/cs-l.gif" width="190" height="46" alt="(x+b/2a)^2 = -c/a + (b/2a)^2" /></sup></b></td>
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</tr>
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<tr>
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<td>Square root</td>
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<td><b><img src="images/latex/dqa-a.gif" width="193" height="51" alt="(x+b/2a) = (+-) sqrt(-c/a+(b/2a)^2)" /></b></td>
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</tr>
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<tr>
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<td>Move b/2a to right</td>
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<td><img src="images/latex/dqa-b.gif" width="193" height="51" alt="x = -b/2a (+-) sqrt(-c/a+(b/2a)^2)" /></td>
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</tr>
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<tr>
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<td colspan="2" align="center"> <i><br />
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That is actually solved! But let's simplify it a bit:<br />
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<br />
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</i> </td>
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</tr>
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<tr>
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<td>Multiply right by 2a/2a</td>
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<td><img src="images/latex/dqa-c.gif" width="281" height="52" alt="x = [ -b (+-) sqrt(-(2a)^2 c/a + (2a)^2(b/2a)^2) ] / 2a" /></td>
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</tr>
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<tr>
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<td>Simplify:</td>
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<td><img src="images/latex/dqa-d.gif" width="158" height="39" alt="x = [ -b (+-) sqrt(-4ac + b^2) ] / 2a" /></td>
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</tr>
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</table>
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<p><br />
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Which is the Quadratic formula we all know and love: </p>
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<p class="center"><img src="images/quadratic-formula.svg" alt="Quadratic Formula: x = [ -b (+-) sqrt(b^2 - 4ac) ] / 2a" /></p>
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<p> </p>
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<div class="related"> <a href="quadratic-equation.html">Quadratic Equations</a> <a href="factoring-quadratics.html">Factoring Quadratics</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-solver.html">Quadratic Equation Solver</a> <a href="index.html">Algebra Index</a> </div>
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