Quadratic Equation Calculator

What this calculator does

Enter the coefficients a, b and c from a quadratic equation written as ax² + bx + c = 0, and this calculator solves for x using the quadratic formula, showing both roots and whether they're real or complex.

The formula

x = (−b ± √(b² − 4ac)) / (2a)

The expression under the square root, b² − 4ac, is called the discriminant. Its sign determines what kind of roots the equation has:

• Discriminant > 0: two distinct real roots
• Discriminant = 0: exactly one real root (a repeated root)
• Discriminant < 0: two complex roots (no real solution)

Worked example

For x² − 3x + 2 = 0 (a=1, b=−3, c=2): discriminant = (−3)² − 4×1×2 = 9 − 8 = 1, which is positive, so there are two real roots. x = (3 ± 1) / 2, giving x = 2 and x = 1. You can check this by substitution: 2² − 3×2 + 2 = 4 − 6 + 2 = 0, correct.

When roots are complex

For x² + 2x + 5 = 0, the discriminant is 4 − 20 = −16, which is negative. Since you can't take the square root of a negative number within real numbers, the roots are complex: x = −1 ± 2i, where i is the imaginary unit (√−1). These solutions are still mathematically valid — they just don't correspond to points where the parabola crosses the x-axis on a standard graph.

Common mistakes

• Forgetting the ± sign — a quadratic has up to two solutions, not one, and missing the second one is a common exam mistake.
• Getting the sign of b wrong when reading off coefficients — for x² − 3x + 2, b is −3, not 3.
• Assuming a negative discriminant means "no answer" rather than "no real answer" — the equation still has complex solutions.

Frequently asked questions