In deriving his formula, Cardano arrives at the equation $y^3+py+q=0$. By substituting $y=\sqrt[3]{u}+\sqrt[3]{v}$, he gets the equation $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +p)=0$. Then he imposes that $u+v+q=0$ and $3\sqrt[3]{u} \sqrt[3]{v} +p=0$.
He "imposes".
My question: Why could he impose? Why isn't there any possible $(u, v)$ out there such that $u+v+q\neq0$ and $3\sqrt[3]{u} \sqrt[3]{v} +p\neq0$ but still $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +p)=0$?
Please help if you could-many thanks in advance!
Think of it in words like this: "Let's suppose that $y$ is a sum of two numbers whose cubes sum to $-p$ and whose product is $-\frac{p}{3}$. Can we find those two numbers?" That's all that is going on.
More formally the $u$ and $v$ are free variables that have been introduced into the problem, in order to aid in solving for $y$. So one is free to impose any conditions on them that one wants to impose; hopefully, with a sufficiently clever set of conditions, one can solve for those free variables and from that compute $y$.