Quadratic Formula

How Did We Get Here

\begin{aligned} a x^{2} + b x + c & = 0 \\ convert to vertex form \\ factor out a \\ a (x^{2} + \frac{b}{a} x) + c & = 0 \\ complete the square \\ {(\frac{\frac{b}{a}}{2})}^{2} \\ = {(\frac{b}{2 a})}^{2} \\ = \frac{b^{2}}{4 a^{2}} \\ add and subtract this \\ a (x^{2} + \frac{b}{a} x + \frac{b^{2}}{4 a^{2}} - \frac{b^{2}}{4 a^{2}}) + c & = 0 \\ factor \\ a {(x + \frac{b}{2 a})}^{2} + a (- \frac{b^{2}}{4 a^{2}}) + c & = 0 \\ a {(x + \frac{b}{2 a})}^{2} - \frac{b^{2}}{4 a} + c & = 0 \\ a {(x + \frac{b}{2 a})}^{2} & = \frac{b^{2}}{4 a} - c \\ multiply and divide c by 4a to get common denominators \\ a {(x + \frac{b}{2 a})}^{2} & = \frac{b^{2}}{4 a} - \frac{4 a c}{4 a} \\ a {(x + \frac{b}{2 a})}^{2} & = \frac{b^{2} - 4 a c}{4 a} \\ {(x + \frac{b}{2 a})}^{2} & = \frac{b^{2} - 4 a c}{4 a^{2}} \\ x + \frac{b}{2 a} & = \frac{\pm \sqrt{b^{2} - 4 a c}}{2 a} \\ x & = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{aligned}