Cubic Equation Calculator

This cubic equation calculator solves any equation of the form a times x cubed, plus b times x squared, plus c times x, plus d, equals zero, working out all three roots in one step. Cubic equations turn up across algebra, physics and engineering whenever a variable is raised to the third power, and finding the roots by hand with Cardano's formula is slow and easy to get wrong, especially when the answers involve complex numbers. You enter the four coefficients, a, b, c and d, with a required to be non-zero, since a zero leading coefficient turns the equation into a quadratic. The calculator returns all three roots together, however many are real and however many form a complex conjugate pair, plus a count of how many roots are real and a repeat of the leading coefficient for reference. Every cubic with real coefficients has at least one real root, and either three real roots or one real root paired with two complex ones, so the result always shows at least one real value. Roots are found numerically using the Durand-Kerner method, which converges on every root at once rather than solving for each separately. Try the worked example below, coefficients 1, -6, 11 and -6, which factors into roots of 1, 2 and 3, to see how the calculator lays out its answer before entering your own numbers.

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