Formula and example
How it is calculated
With A=b/a, B=c/a, C=d/a and x=t−A/3, the equation becomes t³+pt+q=0, where p=B−A²/3 and q=2A³/27−AB/3+C.
Worked example
x³ − 6x² + 11x − 6 = 0 has roots 1, 2 and 3. Substituting x=t+2 gives t³−t=0.
Things to check
- Cardano and trigonometric formulas give numerical approximations. Nearly coincident roots are sensitive to rounding.
- If a = 0, the quadratic or linear calculation takes over.
Frequently asked questions
Why can a root occur more than once?
Multiplicity describes how often its factor occurs in the polynomial. (x−1)³ has only one distinct root, but the root 1 has multiplicity three.