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.