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.