So I am writing an essay on different ways to solve cubics. But I get stuck in the Cardano's method...
Mainly is the part with Cardano's method's condition $\frac{q^2}{4} + \frac{p^3}{27}$ must be bigger than $0.$
What if the condition was not true?
What if the condition was $0$ or negative?
What would happen? Would that make the method more complicated ?
Also, what are the pros and cons, or limitations of this method.
$\frac{q^2}{4}+\frac{p^3}{27}$ need not be bigger than $0$. If it is less than $0$, it has no square root in the real numbers, but it has a square root in the complex numbers. Do you know about those?
Historically, the complex numbers were invented for precisely this reason. The fascinating thing about cubics is that even when a cubic has real coefficients and all three roots are real, you might still need to introduce complex numbers to give a solution by radicals.