Can a discriminant be a fraction?

Yes. If a, b, or c is fractional, b squared minus 4ac can be fractional. Its sign still classifies the roots in the same way.

Discriminant Calculator — Classify Quadratic Roots

Enter a, b, and c from ax² + bx + c = 0. The calculator evaluates Δ = b² − 4ac exactly in your browser; coefficients are not uploaded.

Instantly see whether the quadratic has two distinct real roots, one repeated real root, or two complex-conjugate roots—and whether real roots are rational.

Enter quadratic coefficients

Root classification

Discriminant

Δ = 1

Conclusion

Two distinct real, rational roots

Because Δ is positive and its square root is rational.

Substitution(−5)² − 4(1)(6)

Root count2 distinct real roots

Rootsx = 3 or x = 2

Decimal roots3, 2

Graph meaningThe parabola crosses the x-axis twice.

Discriminant steps

What the discriminant tells you

For ax² + bx + c = 0, with a ≠ 0, the discriminant is the expression beneath the square-root sign in the quadratic formula.

Δ = b² − 4ac and x = (−b ± √Δ)/(2a)

Value of Δ	Roots over the real numbers	Graph of y = ax² + bx + c
Δ > 0	Two distinct real roots	Crosses the x-axis twice
Δ = 0	One repeated real root	Touches the x-axis once at the vertex
Δ < 0	No real roots; two complex roots	Does not meet the x-axis

Rational versus irrational real roots

When a, b, and c are rational, a nonnegative discriminant has a rational square root only when it is a perfect square of a rational number. Then the roots are rational. If Δ > 0 but √Δ is irrational, both real roots are irrational.

Discriminant examples

Two rational roots

For x²−5x+6=0, Δ=(−5)²−4(1)(6)=1. Since 1 is positive and a perfect square, the roots are distinct, real, and rational.

Two irrational roots

For x²+2x−1=0, Δ=2²−4(1)(−1)=8. It is positive but not a perfect square, so the two real roots are irrational.

Repeated root

For x²−6x+9=0, Δ=36−36=0. The parabola touches the x-axis at the repeated root x=3.

Complex roots

For x²+4x+8=0, Δ=16−32=−16. There are no real x-intercepts and the roots are −2±2i.

Accuracy, privacy, and limits

Integers, finite decimals, and fractions are reduced and calculated with exact BigInt fraction arithmetic. Decimal root values are rounded to 10 significant digits. All calculation happens locally in the browser, and the calculator does not store or transmit coefficient inputs.

Each typed number is limited to 18 digits (excluding punctuation and leading zeros) to keep calculations responsive. Scientific notation and repeating decimals are not accepted. Last reviewed: 4 August 2026.

Discriminant calculator FAQs

What is the discriminant of a quadratic?

For ax²+bx+c=0, the discriminant is Δ=b²−4ac. It is the quantity under the square root in the quadratic formula.

How does the discriminant classify roots?

A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives two non-real complex-conjugate roots.

When are quadratic roots rational?

For rational coefficients, real roots are rational exactly when the discriminant is a square of a rational number. A positive nonsquare discriminant produces two irrational real roots.

Can the discriminant be a fraction?

Yes. Fractional or decimal coefficients can produce a fractional discriminant. Its sign still classifies the roots normally.

What if a equals zero?

The expression is no longer quadratic, so this classification does not apply. Use the Linear Equation Solver when a is zero and b is not.

Does a negative discriminant mean there are no roots?

There are no real roots, but there are two roots over the complex numbers. They form a complex-conjugate pair.