A group G is solvable if there exist groups G_0, G_1, . . ., G_k such that:
  1. G_0 is the trivial group.
  2. G_k is G.
  3. G_i is a normal subgroup of G_{i+1}, for 0 <= i < k.
  4. G_{i+1} / G_i is abelian for 0 <= i < k.
An important theorem of Galois theory is that a polynomial can be solved by radicals iff its Galois group is solvable. One can construct a quintic polynomial with Galois group S_5---said group being unsolvable[1]---so there can be no general solution by radicals for polynomials of degree five or higher. That is, there is no analogue of the quadratic formula for polynomials of degree >= 5.

[1]: S_5's only proper normal subgroups are e (the trivial group) and A_5. S_5 / e = S_5 is not abelian, so that doesn't work. S _5 / A_5 = Z_2 is abelian, but A_5, being simple and nonabelian, is not itself solvable. Hence S_5 cannot be solvable.