What is the discriminant?

The discriminant is the expression Δ = b²−4ac that appears under the square root in the quadratic formula x = (−b ± √Δ) / 2a. Its sign tells you how many real roots the equation has without needing to solve fully.

What does a positive discriminant mean?

When Δ > 0, the square root is a real positive number, so the ± produces two distinct real values. The equation has two different real roots.

What does a zero discriminant mean?

When Δ = 0, the square root is zero, so both roots collapse to the single value x = −b / 2a. This is called a repeated or double root.

What does a negative discriminant mean?

When Δ < 0, the square root of a negative number is imaginary, so the two roots are complex conjugates of the form (−b ± i√|Δ|) / 2a. There are no real solutions.

Can I enter decimal or negative coefficients?

Yes. Any real number is accepted for a, b, and c — including negatives, decimals, and zero for b or c. Only a = 0 is invalid because it makes the equation linear, not quadratic.

Does this calculator give me the actual roots?

This tool focuses solely on the discriminant and root classification. For the full roots, use the Quadratic Formula Calculator.

Discriminant Calculator

Name: Discriminant Calculator
Availability: InStock
Author: Nham Vu

Enter coefficients a, b, and c to instantly compute the discriminant b²−4ac and determine whether the quadratic has two real, one real, or two complex roots.

Quadratic Coefficients

Enter the values for the equation ax² + bx + c = 0

a x² + b x + c = 0

a (coefficient of x²)

b (coefficient of x)

c (constant term)

Discriminant Δ = b² − 4ac

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Nature of Roots

Step-by-Step Working

Summary

Enter coefficients a, b, and c to instantly compute the discriminant b²−4ac and determine whether the quadratic has two real, one real, or two complex roots.

How it works

Enter coefficient a (the x² term) — must not be zero.
Enter coefficient b (the x term) and coefficient c (the constant).
The calculator substitutes your values into Δ = b²−4ac.
The numerical value of the discriminant is displayed.
The nature of the roots is classified based on the sign of Δ.
A plain-language explanation of what the result means is shown.

Use cases

Checking whether a quadratic equation has real solutions before solving it.
Algebra homework and exam preparation.
Determining the number of x-intercepts a parabola has.
Verifying discriminants computed by hand.
Teaching the relationship between the discriminant and the nature of roots.
Pre-screening equations in physics and engineering problem sets.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu