Intuition behind the Substitutions for the Derivation of the Cubic Equation

Question

I have seen the derivation of the cubic equation, and it is amazing. There were 2 specific substitutions needed to solve it. From $x^3+ax^2+bx+c=0$ the first substitution is made $y=x+a/3$, and then $y=\sqrt[3]{u}-\sqrt[3]{v}$. My question is, how did people know how to make those substitutions? I see no intuitive way whatsoever to make these substitutions. Did people just use trial and error? Thanks!