The Quadratic Formula

Variables

Symbol	Meaning	Constraints / unit
a	Coefficient of the quadratic term	a ≠ 0
b	Coefficient of the linear term	any real or complex number
c	Constant term	any real or complex number
x	The roots of the equation	two values, with multiplicity

Derivation — completing the square

Divide the equation through by a, then move the constant across:

Add to both sides so the left side becomes a perfect square:

Take square roots of both sides and solve for x:

The discriminant

The quantity decides the character of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives a conjugate pair of complex roots.

Worked example

Solve . Here a = 1, b = −5, c = 6, so Δ = 25 − 24 = 1 and: