Cubic Discriminant Calculator

This calculator finds the discriminant of a cubic equation ax³ + bx² + cx + d = 0. Enter the four coefficients; you get Δ, whose sign tells whether the cubic has three distinct real roots, a repeated root, or one real root and two complex ones.

Formula

Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d².

• Δ > 0: three distinct real roots
• Δ = 0: a repeated root (all roots real)
• Δ < 0: one real root and two complex conjugate roots

Worked Example

Take x³ − 6x² + 11x − 6 = 0 (a = 1, b = −6, c = 11, d = −6).

1. 18abcd = 18 × 1 × (−6) × 11 × (−6) = 7,128
2. −4b³d = −4 × (−216) × (−6) = −5,184
3. b²c² = 36 × 121 = 4,356
4. −4ac³ = −4 × 1,331 = −5,324
5. −27a²d² = −27 × 36 = −972
6. Δ = 7,128 − 5,184 + 4,356 − 5,324 − 972 = 4.0000

How to read the result

Δ = 4 is positive, so the cubic has three distinct real roots; they are 1, 2 and 3, since the cubic is (x − 1)(x − 2)(x − 3). The size of Δ is not meaningful on its own: multiplying every coefficient by k multiplies Δ by k⁴.

Limits and common mistakes

• a must not be 0, or the equation is not a cubic.
• Watch the signs of b and d when cubing and squaring.
• With decimal coefficients, a Δ that should be 0 may show a tiny non-zero number from rounding.

Frequently Asked Questions

Does Δ > 0 always mean three real roots?

Yes, for real coefficients, three distinct real roots.

What if Δ = 0?

At least two roots are equal and all roots are real.

How do I find the roots themselves?

Try rational roots ±(factors of d) ÷ (factors of a), then divide and solve the quadratic, or use Cardano’s formula.

Is this related to the quadratic discriminant?

Yes. Both are products of squared root differences, scaled by the leading coefficient.

in a | a (x³ coefficient) | - | 1 | -
in b | b (x² coefficient) | - | -6 | -
in c | c (x coefficient) | - | 11 | -
in d | d (constant) | - | -6 | -
out disc | Discriminant Δ | - | 18 * a * b * c * d - 4 * b^3 * d + b^2 * c^2 - 4 * a * c^3 - 27 * a^2 * d^2 | 4