A quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. For example,
Finding the zeroes of a polynomial—values of x which satisfy such an equation—given its coefficients was long a prominent mathematical problem. The linear and quadratic cases fell fairly quickly; after awhile cubic and quartic succumbed. But if there was some pattern to the formulæ none could see it, and the quintic was proving to be extremely stubborn.
Eventually Paolo Ruffini and Niels Abel were able to prove that there is no quintic formula. This is somewhat surprising; even though the zeroes exist, there is no single finite expression of +, -, ×, ÷, and radicals that can produce them from the coefficients for all quintics. (One can resort to infinite expressions; Newton's method provides one. See also ‘limit of a sequence’.)
But their proof did not generalise to higher degrees. The honour of proving the quartic formula to be the last of its kind, ie there was no sextic, septic, octic, formula, and so on, fell to Evariste Galois, who had an ingenious insight which reduced the issue to an important but solved question of group theory.