Polynomial Root Finder
Find real roots of a cubic ax³+bx²+cx+d via Cardano's formula plus the discriminant
About this calculator
The Polynomial Root Finder computes the real roots of a cubic ax³ + bx² + cx + d = 0 by depressing the cubic and applying Cardano's method: three distinct real roots come from the trigonometric formula, a zero discriminant yields the repeated-root closed forms, and a negative discriminant gives the single real root via cube roots. It reports the discriminant and the number of distinct real roots, and when a = 0 it gracefully falls back to quadratic or linear solving so lower-degree inputs still work.
How to use the Polynomial Root Finder 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 → three real roots 1, 2, 3 (discriminant 4 > 0)
- x³ − 8 → one real root x = 2 (discriminant −1728, two roots are complex)
- x³ − 3x² + 3x − 1 → repeated root x = 1 (discriminant 0)
- a = 0 falls back to the quadratic x² − 5x + 6 → roots 2 and 3
Frequently asked questions
How does the Polynomial Root Finder 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 Polynomial Root Finder 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.