Discriminant Calculator

Compute the discriminant of a quadratic equation to determine the nature and values of its roots.

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Quadratic equation

Enter coefficients for ax² + bx + c = 0

Coefficient a (x² term) Coefficient b (x term) Coefficient c (constant)

Discriminant

Δ = 1

x² − 3x + 2 = 0

Discriminant (b² − 4ac): 1
Root Nature: Two distinct real roots
Root x₁: 2
Root x₂: 1
Vertex: (1.5, -0.25)

Interpretation

The discriminant is positive (1), so the quadratic equation has two distinct real roots. The parabola crosses the x-axis at two points. The roots are x = 2 and x = 1.

Algebra

Discriminant calculator: determine the nature of quadratic roots

A discriminant calculator computes Δ = b² − 4ac for a quadratic equation ax² + bx + c = 0 and uses the result to classify the roots. The sign of the discriminant tells you whether the equation has two distinct real roots, one repeated real root, or two complex conjugate roots.

Interpreting the discriminant

The discriminant Δ = b² − 4ac appears under the square root in the quadratic formula: x = (−b ± √Δ) / (2a). Its sign determines what lies under that radical.

When Δ > 0, the square root is real and positive, giving two distinct real roots. When Δ = 0, the square root vanishes, giving one repeated root x = −b/(2a). When Δ < 0, the square root of a negative number produces complex conjugate roots.

Δ = b² − 4ac

The discriminant of a quadratic equation.

x = (−b ± √Δ) / (2a)

Quadratic formula using the discriminant.

Frequently asked questions

Can the discriminant be used for higher-degree polynomials?

Yes, discriminants exist for cubic and higher-degree polynomials, but they are more complex. For a cubic ax³ + bx² + cx + d, the discriminant involves all four coefficients. This calculator focuses on the quadratic case.

What does a perfect square discriminant mean?

If the discriminant is a perfect square (like 4, 9, 16, etc.) and the coefficients are integers, the roots are rational numbers. This means the quadratic can be factored over the rationals.

Discriminant Calculator

Δ = 1

Discriminant calculator: determine the nature of quadratic roots

Interpreting the discriminant

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