Cardanos equation is complete, but only under the assumption that complex numbers exist, however, the same is true for the formula of closed-form solutions of quadratic equations. The difference is that Cardanos equation might make use of complex numbers but in the end, returns a real solution whereas the quadratic solver remains complex whenever D < 0.

So in order to implement qubic equations you'd need (a) support for complex numbers, even though the final solution might be real and (b) support for reducing (square and higher order) roots. (b) would be a very valuable addition, as it also allows you to incorporate equations like x^2 - 2 = 0 which solves to x = sqrt(2) or x = -sqrt(2).