Quadratic Formula Calculator

The Discriminant Decides the Root Type

For ax squared + bx + c = 0, the quadratic formula x = (-b plus-or-minus square root of (b squared - 4ac)) / 2a depends entirely on the discriminant, delta = b squared - 4ac. A positive delta gives two distinct real roots, delta = 0 gives one repeated real root, and a negative delta gives two complex roots (no real solutions).

Worked Example

Solve 2x squared - 3x - 5 = 0, so a = 2, b = -3, c = -5. Discriminant: delta = (-3) squared - 4(2)(-5) = 9 + 40 = 49. Since delta is positive and a perfect square, the roots are real and rational: x = (3 plus-or-minus square root of 49) / 4 = (3 plus-or-minus 7) / 4. That gives x1 = 10/4 = 2.5 and x2 = -4/4 = -1. Checking x = -1: 2(-1) squared - 3(-1) - 5 = 2 + 3 - 5 = 0, confirmed.

When Roots Are Complex

If you instead entered a = 1, b = 2, c = 5, the discriminant would be 4 - 20 = -16 (negative). The tool then reports x = -1 plus-or-minus 2i, using i = square root of -1, since a negative number has no real square root.

Why a Can't Be Zero

Setting a = 0 removes the x squared term entirely, turning the equation linear (bx + c = 0) rather than quadratic - the tool rejects this input rather than dividing by zero in the formula's denominator (2a).