Quintic Equation 

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.