>>639
Why are you complitated your life?
There are at least two different easy ways to solve \(x^3+px+q=0\)
One of these is 
\(x=y+\frac{a}{y}\)
\( y^3+\frac{a^3}{y^3}+(3a+p)\left(y+\frac{a}{y}\right)\)+q=0
\(\begin{cases} 3a+p=0\Rightarrow a=-\frac{p}{3}
\\y^6+qy^3-\frac{p^3}{27}=0 \end{cases}\)
\(y^3=-\frac{q}{2}\mp \sqrt{\frac{q^2}{4}+ \frac{p^3}{27}} \)
\(\boxed{x=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+ \frac{p^3}{27}}}+\frac{-\frac{p}{3}}{\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+ \frac{p^3}{27}}}}}\)