C1 Quadratic Formula Proof - Revision history

Ian at 11:39, 1 July 2014

2014-07-01T11:39:54Z

2014-07-01T09:38:34Z

2014-07-01T09:37:05Z

2014-07-01T09:35:57Z

2014-07-01T09:33:20Z

2014-07-01T09:32:22Z

2014-07-01T09:30:07Z

2014-07-01T09:29:52Z

2014-07-01T09:28:59Z

2014-07-01T09:28:33Z

@@ Line 6: / Line 6: @@
 x^2+\frac{b}{a}x+\frac{c}{a}&=0\\
 \text{Complete the square}&&
-\left(x+\frac{1}{2}\frac{b}{a}x\right)^2-\left(\frac{1}{2}\frac{b}{a}\right)^2+\frac{c}{a}&=0\\
+\left(x+\frac{1}{2}\frac{b}{a}\right)^2-\left(\frac{1}{2}\frac{b}{a}\right)^2+\frac{c}{a}&=0\\
 \text{Simplify}&&
-\left(x+\frac{b}{2a}x\right)^2-\frac{b^2}{4a^2}+\frac{c}{a}&=0\\
+\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a^2}+\frac{c}{a}&=0\\
 \text{Rearrange}&&
 \left(x+\frac{b}{2a}\right)^2&=\frac{b^2}{4a^2}-\frac{c}{a}\\

@@ Line 7: / Line 7: @@
 \text{Complete the square}&&
 \left(x+\frac{1}{2}\frac{b}{a}x\right)^2-\left(\frac{1}{2}\frac{b}{a}\right)^2+\frac{c}{a}&=0\\
 \text{Rearrange}&&
 \left(x+\frac{b}{2a}\right)^2&=\frac{b^2}{4a^2}-\frac{c}{a}\\

@@ Line 6: / Line 6: @@
 x^2+\frac{b}{a}x+\frac{c}{a}&=0\\
 \text{Complete the square}&&
-(x+\frac{1}{2}\frac{b}{a}x)^2-(\frac{1}{2}\frac{b}{a})^2+\frac{c}{a}&=0\\
+\left(x+\frac{1}{2}\frac{b}{a}x\right)^2-\left(\frac{1}{2}\frac{b}{a}\right)^2+\frac{c}{a}&=0\\
 \text{Rearrange}&&
-(x+\frac{b}{2a})^2&=\frac{b^2}{4a^2}-\frac{c}{a}\\
+\left(x+\frac{b}{2a}\right)^2&=\frac{b^2}{4a^2}-\frac{c}{a}\\
 \text{Combine fractions}&&
-(x+\frac{b}{2a})^2&=\frac{b^2-4ac}{4a^2}\\
+\left(x+\frac{b}{2a}\right)^2&=\frac{b^2-4ac}{4a^2}\\
 \text{Square root}&&
 x+\frac{b}{2a}&=\pm\sqrt{\frac{b^2-4ac}{4a^2}}\\

@@ Line 1: / Line 1: @@
 <math>
 \begin{aligned}
-\text{The general quadratic}&&ax^2+bx+c&=0\\
+\text{The general quadratic}&&
-x^2+\frac{b}{a}x+\frac{c}{a}&=0&&\text{divide through by }a\\
+ax^2+bx+c&=0\\
 (x+\frac{1}{2}\frac{b}{a}x)^2-(\frac{1}{2}\frac{b}{a})^2+\frac{c}{a}&=0\\
 (x+\frac{b}{2a})^2&=\frac{b^2}{4a^2}-\frac{c}{a}\\
 (x+\frac{b}{2a})^2&=\frac{b^2-4ac}{4a^2}\\
 x+\frac{b}{2a}&=\pm\sqrt{\frac{b^2-4ac}{4a^2}}\\
 x+\frac{b}{2a}&=\frac{\pm\sqrt{b^2-4ac}}{2a}\\
 x&=\frac{\pm\sqrt{b^2-4ac}}{2a}-\frac{b}{2a}\\
 x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
 \end{aligned}</math>

@@ Line 1: / Line 1: @@
 <math>
 \begin{aligned}
-ax^2+bx+c&=0&&\text{the general quadratic}\\
+\text{The general quadratic}&&ax^2+bx+c&=0\\
 x^2+\frac{b}{a}x+\frac{c}{a}&=0&&\text{divide through by }a\\
 (x+\frac{1}{2}\frac{b}{a}x)^2-(\frac{1}{2}\frac{b}{a})^2+\frac{c}{a}&=0\\

@@ Line 9: / Line 9: @@
 x+\frac{b}{2a}&=\frac{\pm\sqrt{b^2-4ac}{2a}}\\
 x&=\frac{\pm\sqrt{b^2-4ac}{2a}}-\frac{b}{2a}\\
-x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}\\
+x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
 \end{aligned}</math>