Cubic Equation Solver

How the cubic equation solver works

Cubic equations can have one real root or three real roots, and that distinction is easy to miss when trial factoring fails. This solver normalizes the leading coefficient, shifts the polynomial into depressed-cubic form, and evaluates its discriminant term. It then uses cube roots for the one-root case or the trigonometric branch for three real roots, returning sorted values for easier checking.

The default coefficients describe x cubed minus 6x squared plus 11x minus 6. Its depressed form has a negative discriminant term, so three real roots are expected. The verified output is 1, 2, and 3. Multiplying x minus 1, x minus 2, and x minus 3 expands back to the entered coefficients and confirms all three solutions.

Reading cubic equation solver results privately

Only real roots are displayed; when a cubic has one real root, the other two complex roots are outside this page’s scope. Coefficients are handled with floating-point arithmetic, and nearly repeated roots may show small rounding noise. The leading cubic coefficient must be nonzero. For symbolic radicals or exact algebraic-number output, use a computer algebra system rather than decimal root estimates. This cubic equation solver calculation runs entirely in your browser, so the numbers you enter never leave your device.