quintic equation

A quintic equation is a polynomial equation of the fifth degree. For centuries, mathematicians sought a general formula, using only radicals (like square roots and cube roots), to solve these equations, similar to how quadratic, cubic, and quartic equations could be solved.1234

In 1799, Paolo Ruffini made an attempt to prove that such a general radical solution for the quintic equation was impossible, though his proof had gaps. Niels Henrik Abel later provided a complete proof of this impossibility in 1824. Évariste Galois further developed this by creating a theory that determines which specific quintic equations can be solved by radicals, based on the properties of the group of permutations of the equation's roots.342