Proof of the quadratic formula

The quadratic formula follows from completing the square. Start with ax² + bx + c = 0, where a ≠ 0.

Step 1: Make the coefficient of x² equal to one

Divide every term by a: x² + (b/a)x + c/a = 0. Move the constant: x² + (b/a)x = −c/a.

Step 2: Complete the square

Add (b/2a)² to both sides. The left side factors as a perfect square:

(x + b/2a)² = (b² − 4ac)/(4a²).

Step 3: Take square roots and isolate x

Taking both square roots and subtracting b/2a gives

x = (−b ± √(b² − 4ac))/(2a).

The expression b² − 4ac is the discriminant. For real coefficients, a positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives a complex conjugate pair.