How to Calculate the Discriminant of a Quadratic

This page calculates the discriminant of a quadratic equation ax² + bx + c = 0. Enter a, b and c; you get the discriminant Δ = b² − 4ac and, when Δ is zero or positive, the two real roots from the quadratic formula.

Formula

Δ = b² − 4ac, and the roots are x = (−b ± √Δ) ÷ (2a).

• Δ > 0: two different real roots
• Δ = 0: one repeated real root
• Δ < 0: no real roots (two complex roots); the root lines show Cannot compute

Worked Example

Take x² − 3x + 2 = 0, so a = 1, b = −3, c = 2.

1. Δ = (−3)² − 4 × 1 × 2 = 9 − 8 = 1.0000
2. √Δ = 1
3. x₁ = (3 + 1) ÷ 2 = 2.0000
4. x₂ = (3 − 1) ÷ 2 = 1.0000

How to read the result

Check the roots by substituting them: 2² − 3 × 2 + 2 = 0 and 1² − 3 + 2 = 0. A positive Δ that is a perfect square (with whole a, b, c) means the quadratic factors over the integers, here (x − 1)(x − 2).

Limits and common mistakes

• a must not be 0; with a = 0 the equation is linear.
• Keep the sign of b: −3 squared is +9.
• For complex roots when Δ < 0: x = −b ÷ 2a ± i √(−Δ) ÷ 2a.

Frequently Asked Questions

What does a negative discriminant mean?

There are no real roots: the parabola does not cross the x-axis.

What does a zero discriminant mean?

One repeated root at x = −b ÷ 2a; the parabola touches the x-axis.

Is the discriminant the same as the quadratic formula?

It is the part under the square root in the formula.

Can a, b or c be fractions?

Yes. Enter them as decimals, for example 0.5.

in a | a (x² coefficient) | - | 1 | -
in b | b (x coefficient) | - | -3 | -
in c | c (constant) | - | 2 | -
out d | Discriminant Δ | - | b^2 - 4 * a * c | 4
out x1 | Root x₁ | - | (-b + sqrt(d)) / (2 * a) | 4
out x2 | Root x₂ | - | (-b - sqrt(d)) / (2 * a) | 4