The Cubic Formula

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

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

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