has an article on it. As I recall (dimly), the basic way to solve the equation is
(1) reduce the x-squared term by using x = t - (b/3a). This leaves us with an equation of the form t^3 + pt + q = 0
(2) let t = u + v, and add another condition that 3uv + p = 0. This allows us to find two roots of our new equation, u^3 and v^3.
(3) work back to find the third root, and then back again to get the roots for the original quadratic equation.
It always seemed like more trouble than it was worth to me--it was enough to know it was possible.