Discriminant Calculator

What this calculator does

So, the discriminant is a single number worked out from a polynomial's coefficients that tells you about the nature of its roots without your having to solve it. This tool computes that number, and it is not limited to quadratics: you can choose a second-degree, third-degree, or fourth-degree polynomial and it will handle each.

You pick the degree from a dropdown, and the boxes for the coefficients adjust to match.

How to use it

1. Choose the degree: second, third, or fourth.
2. Enter the coefficients that appear for that degree.
3. Press Calculate.

The tool opens on the fourth degree, so if you want a plain quadratic, switch the dropdown to second and only the a, b, and c boxes remain.

What a discriminant tells you

The discriminant is a shortcut. Instead of solving a polynomial and then looking at its roots, you compute one number and read the character of those roots straight off it. The most useful signal, at every degree, is this: a discriminant of exactly zero means the polynomial has a repeated root, two or more of its roots coinciding. A nonzero discriminant means the roots are all distinct, and its sign tells you how they split between real and complex.

That is why the discriminant is not a niche idea. It hides inside the quadratic formula, under the square root, precisely because its sign is what decides whether the solutions are real or complex.

The quadratic case, b squared minus 4ac

For a quadratic ax² + bx + c, the discriminant is the familiar b² - 4ac, and its sign reads cleanly:

• Positive: two distinct real roots.
• Zero: one repeated real root.
• Negative: a pair of complex roots.

This is the same quantity the quadratic formula calculator computes before it branches, so if you want the actual solutions rather than just their nature, that tool takes over from here.

Going beyond: cubics and quartics

The idea carries up to higher degrees, and this tool goes with it. A cubic, ax³ + bx² + cx + d, has its own discriminant, a longer expression built from the four coefficients. A quartic, with five coefficients, has a longer one still. The tool holds these formulas and evaluates whichever degree you select. The interpretation stays anchored on the same rule: a discriminant of zero flags a repeated root, and the sign of a nonzero discriminant tells you how the roots divide between real and complex pairs. The formulas grow, but the meaning does not change.

A worked example, step by step

Take the quadratic 2x² + 3x + 1, with a = 2, b = 3, c = 1.

• Discriminant: b² - 4ac = 9 - 8 = 1.
• It is positive, so the quadratic has two distinct real roots.

By contrast, x² - 4x + 4 gives 16 - 16 = 0, a discriminant of zero, which flags a repeated root, and indeed that quadratic is (x - 2)² with the single root x = 2.

Questions people ask

What does a positive, zero, or negative discriminant mean?

For a quadratic: positive means two distinct real roots, zero means one repeated real root, and negative means a complex pair. Zero flags a repeated root at any degree.

Does it solve the equation?

No. It reports the discriminant, which tells you the nature of the roots. To find the actual roots of a quadratic, use the quadratic formula calculator.

Can it handle cubics and quartics?

Yes. Choose third or fourth degree from the dropdown and enter the extra coefficients, and it evaluates the discriminant for that polynomial.

Why is the discriminant under the square root in the quadratic formula?

Because its sign is exactly what decides whether the square root is a real number or not, and so whether the solutions are real or complex.

What is the highest degree it supports?

Fourth. Fifth-degree polynomials are not covered, since their discriminant is not implemented in the tool.

References

On the discriminant across degrees. A polynomial discriminant is built from the differences of the roots and vanishes exactly when two roots coincide, and for a quadratic it takes the familiar b squared minus 4ac.

1. Eric W. Weisstein, "Polynomial Discriminant," from MathWorld, a Wolfram resource, on the discriminant of a polynomial and how it detects repeated roots.
2. Eric W. Weisstein, "Quadratic Equation," from MathWorld, a Wolfram resource, on the quadratic discriminant b squared minus 4ac and the nature of the roots.