MathML: Deriving the Quadratic Formula

This page outlines the derivation of the Quadratic Formula.

$\begin{array}{l}We\; take\; a\; quadratic\; equation\; in\; its\; general\; form, and\; solve\; for\; x:\end{array}$

$\begin{array}{l}a{x}^2+bx+c=0\\ a{x}^2+bx=-c\\ x^2+\frac{b}{a}⁤x=\frac{-c}{a}\text{Divide out leading coefficient.}\\ x^2+\frac{b}{a}⁤x+{\left(\frac{b}{2a}\right)}^2=\frac{-c\left(4a\right)}{a\left(4a\right)}+\frac{b^2}{4a^2}\text{Complete the square.}\\ (x+\frac{b}{2a})(x+\frac{b}{2a})=\frac{b^2-4ac}{4a^2}\text{Discriminant revealed.}\\ {(x+\frac{b}{2a})}^2=\frac{b^2-4ac}{4a^2}\\ x+\frac{b}{2a}=\sqrt{\frac{b^2-4ac}{4a^2}}\\ x=\frac{-b}{2a}\pm \left\{C\right\}\sqrt{\frac{b^2-4ac}{4a^2}}\text{There's the vertex formula.}\\ x=\frac{-b\pm \left\{C\right\}\sqrt{b^2-4ac}}{2a}\end{array}$