Discriminant calculator

Discriminant formula

Δ = b² − 4ac for ax² + bx + c = 0

What the discriminant tells you

Δ > 0: Two distinct real roots. The parabola crosses the x-axis at two points.

Δ = 0: One repeated real root (double root). The parabola touches the x-axis at exactly one point.

Δ < 0: No real roots — two complex conjugate roots. The parabola doesn't cross the x-axis.

Connection to the quadratic formula

The quadratic formula is x = (−b ± √Δ)/(2a). When Δ < 0, the √Δ becomes imaginary, giving complex roots.

Perfect square discriminant

If Δ is a perfect square (1, 4, 9, 16...), the roots are rational numbers and the quadratic factors neatly.