Quadratic Formula

The Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

This formula solves any quadratic equation ax² + bx + c = 0 — even when factoring fails or the numbers are messy.

The expression under the radical, b² − 4ac, is called the discriminant (Δ). It tells you how many real solutions exist before you finish the calculation:

Δ > 0 — two distinct real roots | Δ = 0 — one repeated real root | Δ < 0 — no real roots (2 complex roots)

The ± sign gives two solutions: add √Δ for x₁ and subtract √Δ for x₂. Those are the x-intercepts of the parabola y = ax² + bx + c.

The vertex of the parabola is always at x = −b/(2a), exactly halfway between the two roots.

When to Use the Quadratic Formula

Three methods can solve a quadratic — choose wisely:

Factoring — fastest when the equation factors cleanly with integers. Try this first for small coefficients.
Completing the square — great for deriving the vertex form y = a(x−h)² + k and for proofs. More steps, but always works.
Quadratic formula — the all-purpose method. Use it when factoring fails, coefficients are large/ugly, or the discriminant is not a perfect square.

If the discriminant is a perfect square (1, 4, 9, 16, 25 …), the roots are rational — and the equation could have been factored.

Quadratics still feel tangled?