WorldCalcs

Quadratic Formula Calculator

Enter the coefficients a, b and c to solve ax² + bx + c = 0. Shows the discriminant and the real or complex roots.

Reviewed by the WorldCalcs team · Methodology · Last reviewed: June 2026

Result

Discriminant (D): 1

Type: two real

x₁ = 3

x₂ = 2

What is the quadratic formula?

The quadratic formula solves any equation of the form ax² + bx + c = 0, where a is not zero. The solutions are x = (−b ± √(b² − 4ac)) / (2a). The two answers come from the plus and minus of the ± sign, and they are the points where the parabola crosses the x-axis.

The discriminant

The part under the square root, b² − 4ac, is called the discriminant, and it tells you the nature of the roots before you finish. If it is positive there are two different real roots; if it is zero there is one repeated real root; and if it is negative there are two complex roots that come as a conjugate pair.

Example

For x² − 5x + 6 = 0 (a = 1, b = −5, c = 6), the discriminant is (−5)² − 4(1)(6) = 1, so there are two real roots: (5 ± 1) ÷ 2, which gives 3 and 2. For x² + 2x + 5 = 0 the discriminant is −16, so the roots are the complex pair −1 ± 2i.

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Results are estimates and may contain errors — for general information only, not professional advice. Always verify before relying on them. Disclaimer

How to use

Enter the three coefficients a, b and c of your quadratic equation. The calculator computes the discriminant and the roots.

If the discriminant is positive you get two real roots; if zero, one repeated root; if negative, a conjugate pair of complex roots. If a is zero the calculator flags that the equation is linear, not quadratic.

Frequently asked questions

What is the discriminant?+

The discriminant is b² − 4ac, the expression under the square root in the quadratic formula. Its sign tells you whether the roots are two real, one repeated, or two complex.

What does a negative discriminant mean?+

A negative discriminant means the parabola never touches the x-axis, so there are no real solutions — instead there are two complex roots that form a conjugate pair.

Why must a not be zero?+

If a is zero there is no x² term, so the equation is linear (bx + c = 0), not quadratic, and the quadratic formula does not apply.

Can a quadratic have just one solution?+

Yes. When the discriminant is exactly zero the two roots coincide into a single repeated root, and the parabola just touches the x-axis at one point.

What are complex roots?+

Complex roots contain the imaginary unit i, where i² = −1. They appear in conjugate pairs such as −1 + 2i and −1 − 2i when the discriminant is negative.

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