Cubic Equation Calculator

A cubic equation is a polynomial equation whose highest power is three. Its standard form has coefficients a, b, c, and d, with a nonzero leading coefficient. Every cubic has exactly three roots when repeated and complex roots are counted. Depending on its coefficients, those roots may be three distinct real values, a mix of repeated real values, or one real value accompanied by a complex-conjugate pair. This calculator uses the classical ideas behind Cardano's formula. It first divides all coefficients by the leading coefficient to make the polynomial monic. A substitution then removes the squared term and creates a depressed cubic. Two derived quantities, conventionally called p and q, determine a discriminant. A positive discriminant gives one real root and two complex conjugates. A negative discriminant gives three distinct real roots, calculated with a trigonometric form that avoids unnecessary complex intermediate values. A zero discriminant indicates repeated roots. Complex roots are displayed with a real part and an imaginary part followed by i. Because the original coefficients are real, any nonreal roots occur as conjugates: their real parts match and their imaginary parts have opposite signs. Results are rounded for display, so substituting a rounded irrational root back into the polynomial can leave a tiny residual. Exact integer roots appear without trailing decimal zeros. Cubic equations model volume, motion, optimization, intersections, fluid behavior, and many other scientific or engineering relationships. They also arise when factoring higher-degree polynomials and finding eigenvalues of three-dimensional systems. To use the solver correctly, arrange every term on one side of the equality and combine like powers before entering coefficients. Include zero for any missing term. For example, x cubed plus 1 has a equal to 1, b and c equal to 0, and d equal to 1. Multiplying every coefficient by the same nonzero constant does not change the roots. The calculator returns all three solutions so you can factor the polynomial, inspect repeated values, or continue with numerical analysis.