# The Cubic Formula

$x=\frac{-2b+\left(\frac{-1+\sqrt{-3}}{2}{\right)}^{n}\sqrt[3]{4\left(-2{b}^{3}+9abc-27{a}^{2}d+\sqrt{\left(-2{b}^{3}+9abc-27{a}^{2}d{\right)}^{2}-4\left({b}^{2}-3ac{\right)}^{3}}\right)}+\left(\frac{-1-\sqrt{-3}}{2}{\right)}^{n}\sqrt[3]{4\left(-2{b}^{3}+9abc-27{a}^{2}d-\sqrt{\left(-2{b}^{3}+9abc-27{a}^{2}d{\right)}^{2}-4\left({b}^{2}-3ac{\right)}^{3}}\right)}}{6a}$

The cubic formula gives the solutions of $a{x}^{3}+b{x}^{2}+cx+d=0$ for real numbers $a$, $b$, $c$, $d$ with $a\ne 0$.

Directions: Take $n=0$, $1$, $2$. Use real cube roots if possible, and principal roots otherwise.