I was told the discriminant of the cubic is
$$\Delta=-27q^2-4p^3$$
and that $\Delta>0$ means that there are three real roots. Simply put, why is this the discriminant? I ask this because, looking at Cardano's formula, I thought that we want everything inside the square root to be positive to get real roots(just as in quadratic cases).
Namely, $\frac{q^2}{4}+\frac{p^3}{27}>0$. Which is essentially, $27q^2+4p^3>0$. But the discriminant has a minus on it, and I don't see why. Does the derivation involve resultants and whatnot? Will it be rather complex?
I am wondering if there is a simple explanation as to why this is the case. Similarly, for the quartic, quintic discriminants...will I need to go through resultants for them? Or is there a simpler faster way to determine them?