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 TODO