Getting the third degree

We derive and prove formulae for solving cubic equations, with an eye to providing computer implementations. We consider two families of solutions: the formulae of Tartaglia—Cardano and those of Vi`ete. We show that even if the coefficients in the cubic are purely real, and the cubic roots are all real, it is still the case that complex numbers appear, and their role must be appreciated. We prove theorems giving formulae that are valid for cubic equations with complex coefficients, suitable for computer use, and then give simplifications valid for purely real coefficients. These results form the basis of a Maple function which returns solutions to cubics in optimal forms.