Deriving the Quadratic Formula

This page outlines the derivation of the Quadratic Formula.

We take a quadratic equation in its general form, and solve for x:

\begin{matrix} a x^{2} + b x + c & = & 0 \\ a x^{2} + b x & = & - c \\ x^{2} + \frac{b}{a} x & = & \frac{- c}{a} & Divide out leading coefficient. \\ x^{2} + \frac{b}{a} x + {(\frac{b}{2 a})}^{2} & = & \frac{- c (4 a)}{a (4 a)} + \frac{b^{2}}{4 a^{2}} & Complete the square. \\ (x + \frac{b}{2 a}) (x + \frac{b}{2 a}) & = & \frac{b^{2} - 4 a c}{4 a^{2}} & Discriminant revealed. \\ {(x + \frac{b}{2 a})}^{2} & = & \frac{b^{2} - 4 a c}{4 a^{2}} \\ x + \frac{b}{2 a} & = & \sqrt{\frac{b^{2} - 4 a c}{4 a^{2}}} \\ x & = & \frac{- b}{2 a} \pm {C} \sqrt{\frac{b^{2} - 4 a c}{4 a^{2}}} & There's the vertex formula. \\ x & = & \frac{- b \pm {C} \sqrt{b^{2} - 4 a c}}{2 a} \end{matrix}