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.
I think (and hope) that this apparently innocent question will induce a lot of discussions.
As you say, the formal work done for the analytical solutions of cubic and quartic polynomials is of major importance.
Now, from a practical point of view : I am only concerned by numerical methods and I spent most of my life working with cubic equations because of thermodynamics (in oil and gas industry, most of the models are based on so- called cubic equations of state); the very first step of the calculation of any physical property is : find the root(s). One of the key issues is not to miss the roots when more than one does exist; this is crucial not to say more.
Because of limited precision on computers added to the fact that the coefficients are not rational, the analytical solution is not used. In the early 70’s, I produced a technical report showing that, if using quadruple precision (which would be very expensive if compared to double precision), we could loose up to $0.1$ % of the roots. To give you an idea, in a reservoir simulation, solving the cubic is done zillions of times and one of the by-products of the simulation is the decision of exploiting or not the reservoir.
You could be interested by this paper or this one.
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.