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Mathematical Reference

The Cubic Equation Formula

A comprehensive guide to understanding, deriving, and applying the formula for third-degree polynomial equations of the form ax3 + bx2 + cx + d = 0.

1. The General Form

ax3 + bx2 + cx + d = 0, a ≠ 0

Every cubic equation can be written in this standard form. The coefficients a, b, c, and d are real numbers, and a must be nonzero.

Common special cases include:

  • Monic cubic: when a = 1, so x3 + bx2 + cx + d = 0
  • Depressed cubic: when b = 0, so ax3 + cx + d = 0
  • Pure cubic: when b = 0 and c = 0, so ax3 + d = 0

2. Normalization and Depression

Step 1: Normalize

Divide all terms by a to get a monic polynomial:

x3 + (b/a)x2 + (c/a)x + d/a = 0

Step 2: Depress

Substitute x = t - b/(3a) to eliminate the quadratic term:

t3 + pt + q = 0

The depressed coefficients are:

  • p = (3ac - b2) / (3a2)
  • q = (2b3 - 9abc + 27a2d) / (27a3)

This substitution is the key algebraic move behind Cardano's method.

3. Cardano's Formula

The Heart of Cubic Solving
Delta = q2/4 + p3/27
t = cbrt(-q/2 + sqrt(Delta)) + cbrt(-q/2 - sqrt(Delta))
x = t - b/(3a)

Delta < 0

Three distinct real roots. The trigonometric method is usually the cleanest approach.

Delta = 0

Repeated real roots. At least two roots are equal.

Delta > 0

One real root and two complex conjugate roots.

4. The Trigonometric Method

When Delta < 0, Cardano's radical form becomes awkward even though all three roots are real. The trigonometric version avoids that issue:

r = 2 sqrt(-p/3)
theta = (1/3) arccos(-q / (2 sqrt(-p3/27)))
x1 = r cos(theta) - b/(3a)
x2 = r cos(theta + 2pi/3) - b/(3a)
x3 = r cos(theta + 4pi/3) - b/(3a)

5. Worked Example

Solve: x3 - 6x2 + 11x - 6 = 0

1
Identify: a = 1, b = -6, c = 11, d = -6. The cubic is already monic.
2
Depress: p = -1 and q = 0.
3
Discriminant: Delta = -1/27 < 0, so the equation has three real roots.
4
Roots: x1 = 1, x2 = 2, x3 = 3.
OK
Verify: (x - 1)(x - 2)(x - 3) = x3 - 6x2 + 11x - 6.

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Still have questions?

What is the general cubic equation formula?

The general cubic equation is ax^3 + bx^2 + cx + d = 0 where a is not zero. It can be solved using Cardano's formula after converting to the depressed cubic form t^3 + pt + q = 0.

What is the depressed cubic?

A depressed cubic is a cubic equation without the x^2 term, in the form t^3 + pt + q = 0. It is obtained by substituting x = t - b/(3a) into the general form.

How does Cardano's formula work?

Cardano's formula expresses roots using cube roots and the discriminant Delta. It transforms the general cubic into a depressed cubic, then solves that reduced form algebraically.

When does the cubic formula give complex roots?

When Delta is positive, there is one real root and two complex conjugate roots. When Delta is zero, all roots are real with at least two equal. When Delta is negative, all three roots are real and distinct.