Cubic Equation Calculator
The cubic equation calculator uses Cardano formulae to determine the roots of a cubic polynomial. It also plots the graph of the polynomial.
Real roots3, 2, 1three distinct real roots
Equationx^3 - 6x^2 + 11x - 6 = 0
Number of real roots3
Sum of all roots6always −b/a, including complex roots
Product of all roots6always −d/a for a cubic
Checking each root
| Root | Value of the polynomial |
|---|---|
| 3 | 0 |
| 2 | 0 |
| 1 | 0 |
The method
ax³ + bx² + cx + d = 0
Every cubic has a real root
Because odd-degree polynomials run from −∞ to +∞, they must cross zero somewhere. So a cubic has either one real root and a complex conjugate pair, or three real roots. It can never have zero real roots.
The solution here uses the depressed cubic and, where all three roots are real, the trigonometric form — which avoids the complex arithmetic that Cardano’s formula otherwise forces even when the answers are real.