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	<h1 id="quadratic-formula-derivation">Quadratic Formula Derivation</h1>

<p>The standard form of a quadratic equation is:</p>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mi>b</mi><mi>x</mi><mo>&#x0002B;</mo><mi>c</mi><mo>&#x0003D;</mo><mn>0</mn></mrow></math>

<p>Here, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>&#x0002C;</mo><mi>b</mi><mo>&#x0002C;</mo><mi>c</mi><mo>&#x02208;</mo><mi>&#x0211D;</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>&#x02260;</mo><mn>0</mn></mrow></math></p>

<p>We begin by first dividing both sides by the coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi></mrow></math></p>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>&#x027F9;</mi><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>&#x0002B;</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>&#x0003D;</mo><mn>0</mn></mrow></math>

<p>We can rearrange the equation:</p>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>&#x0003D;</mo><mo>&#x02212;</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac></mrow></math>

<p>We can then use the method of completing the square. (<a rel="noopener" target="_blank" href="https://www.mathsisfun.com/algebra/completing-square.html">Maths is Fun</a> has a really good explanation for this technique)</p>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>&#x0002B;</mo><mo stretchy="false">&#x00028;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>&#x0002B;</mo><mo stretchy="false">&#x00028;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup></mrow></math>

<p>On our LHS, we can clearly recognize that it is the expanded form of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">&#x00028;</mo><mi>x</mi><mo>&#x0002B;</mo><mi>d</mi><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup></mrow></math> i.e <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>&#x0002B;</mo><mn>2</mn><mi>x</mi><mi>&#x000B7;</mi><mi>d</mi><mo>&#x0002B;</mo><msup><mi>d</mi><mn>2</mn></msup></mrow></math></p>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>&#x027F9;</mi><mo stretchy="false">&#x00028;</mo><mi>x</mi><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">&#x00029;</mo><mn>2</mn></msup><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>&#x0002B;</mo><mfrac><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>&#x0002B;</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></mrow></math>

<p>Taking the square root of both sides</p>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable displaystyle="true" rowspacing="3pt" columnspacing="0em 2em"><mtr><mtd columnalign="right"><mi>x</mi><mo>&#x0002B;</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd><mtd columnalign="left"><mi /><mo>&#x0003D;</mo><mfrac><mrow><msqrt><mrow><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>&#x0002B;</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mi /><mo>&#x0003D;</mo><mfrac><mrow><mi>&#x000B1;</mi><msqrt><mrow><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>&#x0002B;</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt><mo>&#x02212;</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><mi /><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mi>b</mi><mi>&#x000B1;</mi><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr></mtable></mrow></math>

<p>This gives you the world famous quadratic formula:</p>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>x</mi><mo>&#x0003D;</mo><mfrac><mrow><mo>&#x02212;</mo><mi>b</mi><mi>&#x000B1;</mi><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>&#x02212;</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></math>

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