Cubic Equation Solver
Real and complex roots of a cubic ax³+bx²+cx+d
About this calculator
The Cubic Equation Solver finds all three roots of ax³ + bx² + cx + d = 0 over the complex numbers. It classifies the equation with the exact discriminant Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²: three distinct real roots use the trigonometric method, repeated roots use exact rational closed forms, and the one-real-two-complex case uses Cardano's formula, all in 30-digit decimal arithmetic. Complex roots are returned as conjugate pairs, and a = 0 is rejected with a clear error.
How to use the Cubic Equation Solver calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- x³ − 6x² + 11x − 6 = 0 → x = 1, 2, 3 (Δ = 4, three real roots)
- x³ − 8 = 0 → x = 2 and the conjugate pair −1 ± 1.7320508…i (Δ = −1728)
- x³ − 3x² + 3x − 1 = 0 → triple root x = 1 (Δ = 0)
- x³ + x = 0 → x = 0 and the pure-imaginary pair ±i (Δ = −4)
Frequently asked questions
How does the Cubic Equation Solver calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Cubic Equation Solver calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.