BYTETOOLS

Discriminant Calculator

Compute the discriminant b² − 4ac of a quadratic equation, classify its roots as two real, one repeated or complex, and see the actual root values.

1x² − 3x + 2 = 0

1
Discriminant b² − 4ac
2 real
Nature of roots

Two distinct real roots

D = b² − 4ac = (-3)² − 4 × 1 × 2 = 9 − 8 = 1

  • x₁ = (−b + √D) / 2a = 2
  • x₂ = (−b − √D) / 2a = 1

The parabola crosses the x-axis at two points.

What is the Discriminant Calculator?

The ByteTools Discriminant Calculator evaluates b² − 4ac for any quadratic equation ax² + bx + c = 0 and tells you immediately what kind of roots it has: two distinct real roots when the discriminant is positive, one repeated real root when it is zero, and a complex-conjugate pair when it is negative.

  • Instant discriminant b² − 4ac with the substitution written out
  • Classifies roots: two real, repeated, or complex conjugate pair
  • Computes the actual roots via the quadratic formula
  • Complex roots displayed as a ± bi
  • Explains what the discriminant means graphically
  • 100% private — runs entirely in your browser

How to use the Discriminant Calculator

  1. 1

    Enter the coefficients a, b and c of ax² + bx + c = 0.

  2. 2

    Read the discriminant value b² − 4ac with the substitution shown.

  3. 3

    Check the root classification: two real, one repeated, or two complex.

  4. 4

    Read the actual root values, real or complex.

  5. 5

    Copy the results with one click.

About the Discriminant Calculator

The ByteTools Discriminant Calculator evaluates b² − 4ac for any quadratic equation ax² + bx + c = 0 and tells you immediately what kind of roots it has: two distinct real roots when the discriminant is positive, one repeated real root when it is zero, and a complex-conjugate pair when it is negative.

The tool does not stop at classification — it also computes the roots themselves with the quadratic formula, including the complex case written as a ± bi, and shows the substitution b² − 4ac with your numbers plugged in. That makes it a fast homework checker for algebra students and a clean reference for anyone brushing up.

All calculations run 100% locally in your browser with JavaScript. Coefficients are never uploaded, results update instantly as you type, and everything can be copied with one click.

Frequently asked questions

What does the discriminant tell you?

The discriminant b² − 4ac determines how many real solutions a quadratic has. Positive means two distinct real roots, zero means exactly one repeated real root, and negative means no real roots — instead there is a pair of complex-conjugate roots. It also tells you whether the parabola crosses, touches or misses the x-axis.

How do you calculate the discriminant?

Square the middle coefficient b, then subtract 4 times a times c. For x² − 3x + 2, that is (−3)² − 4(1)(2) = 9 − 8 = 1, which is positive, so there are two real roots (they are 1 and 2). The calculator shows this substitution with your numbers.

What happens when the discriminant is negative?

The square root of a negative number is imaginary, so the quadratic formula produces complex roots of the form a ± bi. For x² + x + 1, the discriminant is 1 − 4 = −3 and the roots are −0.5 ± 0.866i. The parabola never touches the x-axis.

What does a zero discriminant mean for the graph?

It means the parabola touches the x-axis at exactly one point — its vertex sits right on the axis. The single repeated root is x = −b / 2a. For example x² − 4x + 4 has discriminant 0 and the repeated root x = 2.

Is the discriminant only for quadratics?

No — cubics and higher-degree polynomials have discriminants too, with more complicated formulas. But b² − 4ac is by far the most used, because it appears under the square root in the quadratic formula. For cubic classification, see the cubic equation solver.

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