MathML: 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:

a x 2 + b x + c = 0 a x 2 + b x = c x 2 + b a x = c a Divide out leading coefficient. x 2 + b a x + ( b 2 a ) 2 = c ( 4 a ) a ( 4 a ) + b 2 4 a 2 Complete the square. ( x + b 2 a ) ( x + b 2 a ) = b 2 4 a c 4 a 2 Discriminant revealed. ( x + b 2 a ) 2 = b 2 4 a c 4 a 2 x + b 2 a = b 2 4 a c 4 a 2 x = b 2 a ± { C } b 2 4 a c 4 a 2 There's the vertex formula. x = b ± { C } b 2 4 a c 2 a