TITLE: Applications of the formula expressing roots of a general cubic polynomial
QUESTION [8 upvotes]: I know that the mathematics related to finding the general formula by expressing the roots of a third (and fourth) degree polynomial by means of radicals has had an impressive impact on mathematics (complex numbers and group theory, just to say).
However: are there any useful application of these formulas? My impression is that calculus (i.e. computing derivatives, bisection methods, Sturm theorem) are more useful both for qualitative and numerical properties of such roots even if a formula is available.

REPLY [5 votes]: I wouldn't say that the analytical formulae are useless for computation, but they are certainly very much over-rated. When people manage to reduce a problem to solution of a cubic or quartic, you often find that they rejoice and just declare victory. But, in fact, it takes a great deal of care and skill to convert the formulae into code that will work reliably, and it's often easier to just give up and use numerical methods. I work on a large software system where cubic equations often occur (because we often use cubic splines or cubic Bezier curves). We use analytical methods, but the code includes extensive checking to make sure that they don't go wrong, and we often "polish" solutions that we obtain analytically by doing one or two Newton-Raphson iterations, or something similar.
It's not obvious how to code the quadratic formula properly, even, and cubics and quartics are far far worse.
Recommended reading: "Pitfalls in Computation, or Why a Math Book Isn't Enough", by George Forsythe. Available here.