The irreducible case of the cubic, namely the case where Cardan's formula leads to the square root of negative numbers, was studied in detail by Rafael Bombelli in 1572 in his work Algebra.

In the years after Cardan's Ars Magna many mathematicians contributed to the solution of cubic and quartic equations. Vičte, Harriot, Tschirnhaus, Euler, Bezout and Descartes all devised methods.

Thomas Harriot made several contributions. One of the most elementary to us, yet showing a marked improvement in understanding, was the observation that if x = b, x = c, x = d are solutions of a cubic
then the cubic is

Leibniz wrote a letter to Huygens in March 1673. In it he made many contributions to the understanding of cubic equations. Perhaps the most striking is a direct verification of the Cardan-Tartaglia formula. This Leibniz did by reconstructing the cubic from its three roots (as given by the formula) as Harriot claimed in general. Nobody before Leibniz seems to have thought of this direct method of verification. It was the first true algebraic proof of the formula, all previous proofs being geometrical in nature.

The cubic equation is the closed-form solution for the roots of a cubic polynomial. A general cubic equation is of the form