Cubic Equation Solver

Cubic equation: ax³ + bx² + cx + d = 0

A cubic equation ax³ + bx² + cx + d = 0 always has at least one real root, and it can have as many as three. The closed-form answer is Cardano's formula, which appeared in Ars Magna (1545). You start by collapsing the cubic into its depressed form t³ + pt + q = 0 via the substitution x = t − b/(3a). From there, t = ∛(−q/2 + √(q²/4 + p³/27)) + ∛(−q/2 − √(q²/4 + p³/27)). The backstory is messy. Tartaglia worked out the method, handed it over after Cardano swore to keep it quiet, and Cardano printed it regardless. Then there is the awkward part: when the discriminant Δ = q²/4 + p³/27 comes out negative (the casus irreducibilis), the three real roots refuse to be written without taking square roots of negative numbers. That is what pushed mathematicians into their first serious dealings with imaginary numbers, the ones Bombelli would tidy up in 1572.

Applications

Cubics turn up in chemical kinetics (third-order reactions), in solid mechanics (strain-energy and stability problems), in signal processing (cubic transfer functions inside filters), in electronics (the van der Pol oscillator and other cubic nonlinearities), and in thermodynamics (cubic equations of state such as van der Waals). They also show up in exam prep, where the Brazilian ENEM and vestibulares like to hand you a cubic that factors nicely through its rational roots.

FAQ

Can a cubic have no real roots? No. Any cubic with real coefficients has at least one real root, since the function is continuous and runs off to ±∞ at both ends, so it has to cross zero somewhere.

What is the casus irreducibilis? It is the situation where the cubic has three distinct real roots, yet Cardano's formula can only write them using complex numbers. Historically that was the first sign that imaginary numbers couldn't be avoided, even when the answers themselves were perfectly real.

Easier alternative for exam problems? Reach for the rational root theorem. Test the integer divisors of d/a, and once you find one that works, divide by (x − root) to knock the problem down to a quadratic.