Cubic Equation Solver
Solve cubic equations from a full equation or coefficients with Cardano's method, real and complex roots, a depressed-cubic discriminant, and a graph.
A cubic equation is solved here by reducing it to a depressed cubic and applying Cardano's method. The displayed discriminant belongs to that reduced step; its sign classifies the roots.
What this cubic equation solver does
A cubic equation has degree three after simplification. This solver accepts a complete equation or four coefficients, moves all terms to one side, collects the polynomial, and solves the resulting cubic.
x is the unknown; a, b, c, and d are the collected coefficients, with a not equal to zero for a true cubic.
y is the shifted variable used to remove the quadratic term.
p and q are the depressed-cubic parameters used by Cardano's method.
Delta is the depressed-cubic discriminant used here to classify the roots.
- Reduce the equation to a polynomial of degree three or lower
- Shift to a depressed cubic so Cardano's method can be applied
- Use the depressed-cubic discriminant to classify root types
- Convert roots back to the original variable
- Show real roots, complex conjugate roots, and a graph when useful
Root types
The sign of the depressed-cubic discriminant determines whether the solver shows three real roots, a repeated-root case, or one real root with a complex conjugate pair.
| Discriminant sign | Roots | Method |
|---|---|---|
| Positive | Three distinct real roots | Trigonometric form |
| Zero | Repeated roots | Multiple-root special case |
| Negative | One real root and two complex roots | Cardano form with cube roots |
Equation input and coefficient input
Search intent for a cubic equation calculator usually expects both coefficient entry and a complete equation box. This tool supports parentheses, a right-hand side, decimal coefficients, fractions, and implicit multiplication.
- works for products of three linear factors
- accepts equations with a nonzero right-hand side
- tracks simple excluded values when variable denominators can be cleared
- shows an error if simplification creates degree higher than three
Graph and limits
The graph helps check the real roots as x-axis crossings. The solver also reports the inflection point and local extrema when they exist.
Frequently Asked Questions
Sources and References
Calculations are based on the listed reference sources. Links open in a new tab.
Related Tools
Solve one-variable linear equations with steps, fractions, parentheses, domain restrictions, and an answer check.
Solve biquadratic equations with the x-squared substitution method, intermediate values, real or complex roots, and a graph of the even polynomial.
Solve a 2x2 system of linear equations with Cramer's rule, determinants, solution classification, fractions, and substitution check.