Everyday utility

Quadratic Formula Calculator

Enter coefficients a, b, and c to calculate real or complex roots, the discriminant, and the parabola vertex.

Last reviewed May 18, 2026 by ToolSpilo Editorial Team.

Review method: Reviewed against the implemented quadratic formula logic and discriminant examples, displayed formulas, and worked examples.

Calculator tool

How this calculator works

Use the explanation to understand the formula, assumptions, and practical limits behind the calculator result.

The Formula

For a quadratic equation written as ax2+bx+c=0ax^2 + bx + c = 0 with a≠0a \ne 0:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The Discriminant Tells the Story

The part under the square root, b2−4acb^2 - 4ac, is called the discriminant.

DiscriminantWhat it means
Positivetwo real roots
Zeroone repeated real root
Negativetwo complex roots

A Simple Example

For x2−5x+6=0x^2 - 5x + 6 = 0, the discriminant is 25−24=125 - 24 = 1, so there are two real roots. The formula gives 2 and 3.

Why aa Cannot Be Zero

If a=0a = 0, the x2x^2 term disappears and the equation is no longer quadratic. The calculator blocks that case because the quadratic formula would be the wrong tool.

Frequently asked questions

What does the discriminant tell me?

It tells you the root type before you finish solving: two real roots, one repeated real root, or a complex pair.

Why is a = 0 rejected?

Because the x2x^2 term disappears and the equation becomes linear instead of quadratic.

Why are complex roots shown with i?

When the discriminant is negative, the square root uses the imaginary unit i=−1i = \sqrt{-1}.

What does the vertex add?

The vertex shows the parabola's turning point, which helps connect the equation with its graph.