The quadratic formula solves every quadratic equation of the form ax² + bx + c = 0. This guide covers the formula itself, a full derivation from completing the square, how to read the discriminant (b² − 4ac) to predict whether you will get two real roots, a repeated root, or complex roots, when to use the formula versus factoring or completing the square, common mistakes, and several worked examples including projectile motion and revenue optimization.