Discriminant Calculator
Compute the discriminant b² − 4ac of a quadratic ax² + bx + c and see how many real or complex roots it has.
How it works
Δ = b² − 4ac · Δ > 0 → 2 real roots · Δ = 0 → 1 repeated root · Δ < 0 → 2 complex roots
The discriminant is the piece under the square root in the quadratic formula, Δ = b² − 4ac, and its sign alone reveals how many roots a quadratic has before you solve it. A positive discriminant means the parabola crosses the x-axis twice, giving two distinct real roots; zero means it just touches, giving one repeated root at the vertex; negative means it never crosses, so the two roots are complex conjugates. When Δ is a perfect square with integer coefficients, the real roots are rational.
Worked example
For x² − 5x + 6 (a = 1, b = −5, c = 6): Δ = (−5)² − 4·1·6 = 25 − 24 = 1. Since 1 is positive, there are two distinct real roots — and because 1 is a perfect square, they are rational (in fact x = 2 and x = 3). Had c been 7, Δ would be 25 − 28 = −3, signalling two complex roots instead.
Sources: Discriminant — Wikipedia
Frequently asked questions
What does a negative discriminant mean?
The quadratic has no real roots — its parabola never touches the x-axis. The two roots are complex conjugates of the form p ± qi, so the calculator reports two complex roots rather than any real value.
Why does Δ = 0 give only one root?
The ± square-root term in the quadratic formula vanishes when the discriminant is zero, so both roots collapse to the same value, x = −b ÷ 2a. This repeated root is exactly where the parabola’s vertex touches the x-axis.
How does the discriminant tell rational from irrational roots?
With integer coefficients, if Δ is a positive perfect square (like 1, 4, 9) the square root is a whole number and the roots are rational. If Δ is positive but not a perfect square, the roots are real but irrational.
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This calculator is for educational purposes. Double-check important results before acting on them.