Methods for solving quadratic equation were known to many
ancient peoples, including the Babylonian, Chinese and Hindu
civilizations. Whether or not they had the quadratic formula,
they had the main principle (completion of the square). After
much struggle, Scipio del Ferro (1465 -- 1526) found a formula
for solutions of the cubic equation. His work was never
published and was communicated to a few friends. Nonetheless,
the fact that he had solved the problem became known, and
Tartaglia (1500 -- 1557) rediscovered the formula in 1535. He
too kept his result secret but was tricked into revealing it by
Cardano (1501 -- 1576). Cardano also discovered a formula for
the quartic equation.
For 350 years no further progress was made. Finally
Evariste Galois (1811 -- 1832) showed that there is no
algebraic formula for solving equations of degree five
or more. Galois' work relies on the theory of groups.
Note that there are, of course, ways of finding
approximate roots of equations of degree five or more.
e.g., the method of bisection and Newton's method.