I was recently reading about Cardano's method for solving the cubic of the form $t^3 + pt + q = 0 $.
So, you substitute $t$ with two linear variables $t = u+v$. You get the equation $u^3 + v^3 + (3uv +p)(u+v) +q = 0$. Everything is okay thus far. However, then Allan Clark, whose book I was reading, states:
Since we have substituted two variables, $u$ and $v$, in place of the one variable $t$, we are free to require $3uv + p = 0$.
He writes this as if it is the most obvious thing in the world, but I cannot understand why this is true.
So, how do you justify that step?