What does the discriminant tell me?

Δ = b² − 4ac. If Δ > 0 there are two real roots, if Δ = 0 one repeated root, and if Δ < 0 two complex conjugate roots.

When should I factor instead?

If Δ is a perfect square, the roots are rational and the quadratic factors nicely over the integers — factoring is usually faster.

Does it handle decimals?

Yes. Decimal coefficients are converted to exact fractions when possible.

Quadratic Formula Calculator

Q: When should I factor instead?

If Δ is a perfect square, the roots are rational and the quadratic factors nicely over the integers — factoring is usually faster.

Q: Does it handle decimals?

Yes. Decimal coefficients are converted to exact fractions when possible.

Enter the coefficients of ax² + bx + c = 0. The calculator computes the discriminant, explains what it tells you about the roots, applies the quadratic formula exactly (simplifying square roots), and shows factoring or completing the square as an alternative method.

How to use the quadratic formula calculator

Rewrite your equation in standard form ax² + bx + c = 0.
Enter a, b and c (a cannot be zero).
Read the discriminant, the exact roots and their decimal approximations.

Examples

x² − 3x − 10 = 0

Answer: x = −2, x = 5

2x² − 3x − 1 = 0

Answer: x = (3 ± √17)/4

x² + 2x + 5 = 0

Answer: x = −1 ± 2i

Formulas

Quadratic Formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Vertex of a Parabola

h = - \frac{b}{2 a}, k = f (h)