An equation of the form $ax^5+bx^4+cx^3+dx^2+ex+f=0$

It is well known that the quadratic equation has a closed form solution. It is less well known that the cubic equation and quartic equation also both have closed form solutions, although they are significantly more complex.

For centuries a general solution for the quintic (and higher) equation was sought, but in 1824 Abel proved that such a formula (using radicals) is impossible, and this was generalised and extended by Galois's work of 1832 or so.