I've heard of a number of ways that people have derived a cubic formula (I've even heard of a number of different ways to show the formula itself too). What I want to know is how to derive it without any substitutions (like the wikipedia article on it). I know it will be messy, but at least I'll be able to understand it better.
More specifically, I want to know how to solve it without using such complex methods that everyone seems to be using, and don't just link to wolfram (you could link to another step-by-step solver that's free though).
$ax^3+bx^2+cx+d=0$
...
$x=?$