For degree 1, 2, 3 and 4 there is an "extended a,b,c-formula" (like the one we learn in middle or high school, http://en.wikipedia.org/wiki/Quadratic_equation) for the solution to a polynomial equation $p(x) = 0$, when the degree is $\geq 5$ there isn't a solution in this form with the help of radicals due to an application of Galois theory.
My question: how far can you get with this kind of (a,b,c)-formulae when you allow non-radical solutions when solving polynomial equations ?