Cubic Equation Solver
Find all real roots of any cubic equation a·x³ + b·x² + c·x + d = 0 instantly.
How it works
- 1Enter the four coefficients a, b, c, and d for your cubic equation a·x³ + b·x² + c·x + d = 0.
- 2The solver applies Cardano’s method and the trigonometric form for three-root cases to find every real root.
- 3Roots are sorted ascending and de-duplicated, so a repeated root appears only once in the result.
Use cases
- Solve cubic polynomial equations that arise in physics, engineering, and geometry problems.
- Verify roots found by hand or check the output of a computer-algebra system for simple cubics.
- Explore how changing coefficients shifts the roots of a cubic — useful for algebra students.
Frequently asked questions
Can a cubic equation have no real roots?
No — every cubic with real coefficients has at least one real root, because the curve runs from −∞ to +∞ and must cross the x-axis. It may have one or three real roots (counting multiplicity).
What does a repeated root mean here?
A repeated root occurs when the cubic touches the x-axis without crossing it. The solver lists each distinct value once, so a root of multiplicity 2 or 3 still appears as a single entry.
What happens if a = 0?
Setting a to 0 reduces the equation to a quadratic b·x² + c·x + d = 0, which the solver handles automatically using the quadratic formula.
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