I started reading a book of complex analysis and to introduce the complex numbers it gives as example the solution to cubic equation.
Particularly it says that every cubic equation in $x$ can be rearranged in the following form: $$ x^3 = 3px +2q. $$ At this point the solution of such an equation is: $$ x = \sqrt[3]{q + \sqrt{q^2 - p^3}} + \sqrt[3]{q - \sqrt{q^2 - p^3}} . $$ I will skip all the passage but everything works fine and it is indeed the solution (good work Cardano!!!).
The book considers the following equation $$x^3 = 15x+4$$
A solution to this equation is $x=4$ but with the Cardano's formula we must go through complex numbers (because the argument of square root is negative) even to get a real number solution.
However I tried to apply Ruffini's rule and I got in a easy way the solution $x=4$.
My question is how is that that with Cardano's formula we must go through complex number to get a real solution while this does not happen (I assume) with Ruffini's rule?
How do these two methods are different?
Applying Ruffini's rule, you can check if $4$ is or not a solution of the equation; it turns out that it is. But Ruffini's rule doesn't allow you to find a solution for a general cubic.
In the XIXth century, Pierre Wantzel proved that you have to go through complex numbers. To be more precise: if $P(x)$ is a cubic polynomial with rational coefficients, without a rational root, and with $3$ real roots, then every algebraic way of expressing the roots from the coefficients (that is, every expression which uses only sums, subtractions, products, quotients and radicals) must necessarily involve complex non-real numbers. This is the so-called casus irreducibilis of the cubic.