Calculate the root of a cubic equation from coefficients a, b, c, and d using Cardano’s formula, and show the steps and whether roots are real or complex.
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Cardano’s Formula Formula
Cardano’s formula solves a cubic equation in the form:
a x^3 + b x^2 + c x + d = 0
The calculator first converts the cubic to a depressed cubic using these quantities:
f = (3c/a - b^2/a^2)/3
g = (2b^3/a^3 - 9bc/a^2 + 27d/a)/27
h = g^2/4 + f^3/27
If h > 0, the cubic has one real root, and the calculator uses:
x = cubert(-g/2 + sqrt(h)) + cubert(-g/2 - sqrt(h)) - b/(3a)
If h ≤ 0, the cubic has three real roots or repeated real roots. The calculator returns one real root using the trigonometric form:
x = 2sqrt(-f/3)cos((1/3)arccos((-g/2)/sqrt(-f^3/27))) - b/(3a)
When f ≈ 0, the depressed cubic reduces to a simpler form:
x = -cubert(g) - b/(3a)
- a, b, c, d are the coefficients of the cubic equation.
- x is the root returned by the calculator.
- f and g are the depressed cubic coefficients.
- h determines the nature of the roots.
- sqrt means square root.
- cubert means cube root.
The calculator takes your values for a, b, c, and d, checks that a is not zero, calculates f, g, and h, then chooses the matching Cardano method. The result field shows one real root of the cubic equation.
Root Type Based on h
| Condition | Root behavior | Calculator method |
|---|---|---|
| h > 0 | One real root and two complex roots | Real cube-root form |
| h = 0 | Three real roots, with at least two equal | Trigonometric or simplified real-root form |
| h < 0 | Three distinct real roots | Trigonometric form |
Coefficient Setup for Common Cubic Forms
| Cubic equation | a | b | c | d |
|---|---|---|---|---|
| x³ + x + 1 = 0 | 1 | 0 | 1 | 1 |
| 2x³ – 5x² + 3x – 7 = 0 | 2 | -5 | 3 | -7 |
| x³ – 6x² + 11x – 6 = 0 | 1 | -6 | 11 | -6 |
Example Problems
Example 1: One real root
Solve:
x^3 + x + 1 = 0
Here, a = 1, b = 0, c = 1, and d = 1.
f = 1
g = 1
h = 1^2/4 + 1^3/27 = 0.287037
Since h > 0, there is one real root. The calculator returns approximately:
x = -0.682328
Example 2: Three real roots
Solve:
x^3 - 6x^2 + 11x - 6 = 0
Here, a = 1, b = -6, c = 11, and d = -6.
f = -1
g = 0
h = 0^2/4 + (-1)^3/27 = -0.037037
Since h < 0, the equation has three distinct real roots. The calculator returns one of them:
x = 3
FAQ
What equation form should I enter?
Enter the coefficients from a cubic equation written as ax³ + bx² + cx + d = 0. If a term is missing, use zero for that coefficient. For example, x³ + x + 1 = 0 has a = 1, b = 0, c = 1, and d = 1.
Why must a be nonzero?
The coefficient a is the coefficient of the x³ term. If a = 0, the equation is not cubic. It becomes a quadratic, linear, or constant equation, so Cardano’s formula does not apply.
Does Cardano’s formula give all three roots?
Cardano’s formula can be used to find all roots of a cubic, but this calculator displays one real root. When h > 0, that is the only real root. When h ≤ 0, the equation has multiple real roots, and the displayed value is one of them.
