Scientific · Quadratic Equation

Quadratic Equation Solver

Solve ax² + bx + c = 0 for real or complex roots.

Quadratic Equation Solver

Enter a, b, and c (a can't be 0), then select Solve.

How this is solved

Every quadratic equation is solved with the quadratic formula: x = (−b ± √(b² − 4ac)) ⁄ 2a The discriminant (the part under the square root) determines the type of roots: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex roots. A worked breakdown — the discriminant and the rest of the calculation — is shown below the result once you solve.

Examples

x² − 3x + 2 = 0 has two real roots: x = 1 and x = 2. x² + 1 = 0 has no real roots — its solutions are the complex pair x = ±i.

About this calculation

This uses the standard quadratic formula taught in algebra, valid for any real coefficients a, b, and c, with a ≠ 0.

Related tools

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Frequently asked questions

What does it mean if the roots are complex?

It means the equation has no real-number solution — the parabola described by the equation never crosses the x-axis. The complex roots are still mathematically valid answers, expressed as a real part plus or minus an imaginary part.

Why does the calculator say my input is invalid?

The value of a can't be 0 — if it were, the equation would be linear (bx + c = 0), not quadratic.