THE NUMORIX GUIDE
How to use the Cubic Equation Calculator
Last reviewed September 14, 2026
What this calculator does
For ax^3+bx^2+cx+d=0, transform to a depressed cubic t^3+p*t+q=0, calculate Delta=-4p^3-27q^2, and use the corresponding trigonometric or Cardano branch to return real roots.
Formula and method
For ax^3+bx^2+cx+d=0, transform to a depressed cubic t^3+p*t+q=0, calculate Delta=-4p^3-27q^2, and use the corresponding trigonometric or Cardano branch to return real roots.
Variables and inputs
Coefficients a,b,c,d default to 1,-6,11,-6. The leading coefficient a must be nonzero; the UI reports real roots and the discriminant.
Worked example
For x^3-6x^2+11x-6=0, the polynomial factors as (x-1)(x-2)(x-3), so roots are 1, 2, and 3. The transformed p=-1, q=0 gives Delta=4>0, indicating three distinct real roots.
How to interpret the result
The discriminant classifies the real-root pattern: positive here means three distinct real roots, near zero means repeated roots, and negative means one real root plus a complex pair.
Common mistakes to avoid
Enter coefficients in descending powers x^3, x^2, x, constant. Do not interpret the discriminant sign using the quadratic rule without noting this cubic engine's definition.
Assumptions and limitations
When Delta<0, the engine returns only the real root and explicitly omits the complex pair. Floating-point Cardano/trigonometric calculations can be sensitive near repeated roots.
Practical use and checks
The Cubic Equation Calculator solves a*x^3 + b*x^2 + c*x + d = 0 and reports a discriminant-based root set. Enter a = 1, b = -6, c = 11, and d = -6 as a check; the polynomial factors as (x - 1)(x - 2)(x - 3), so the roots should be 1, 2, and 3 and the discriminant should be positive. Substitute each displayed root into the original polynomial to catch rounding or sign mistakes. The discriminant separates three distinct real roots, repeated roots, and one real root with a complex-conjugate pair in the engine's classification. A leading coefficient of zero is not cubic and should be handled by a linear or quadratic solver instead. The page displays only the real root when two roots are complex, so the returned list is not always a complete set of complex solutions. Nearly repeated roots are sensitive to coefficient rounding, and floating-point formulas can produce small errors when the discriminant is close to zero. The calculator does not attach physical meaning, enforce a domain such as x >= 0, or provide symbolic factorizations for every coefficient set. Use residual substitution and an independent solver for critical engineering or statistical models, especially when a root lies near a boundary that controls the decision.