Solve x^3-3wz+sqrt{-1}(ca^3-1/c)=0

$$x ^ { 3 } - 3 w z + \sqrt { - 1 } ( c a ^ { 3 } - \frac { 1 } { c } ) = 0$$

Solve for a

$$a=-c^{-\frac{2}{3}}\sqrt[3]{-icx^{3}+3icwz-1}$$

$$a=e^{\frac{\pi i}{3}}c^{-\frac{2}{3}}\sqrt[3]{-icx^{3}+3icwz-1}$$

$$a=e^{\frac{5\pi i}{3}}c^{-\frac{2}{3}}\sqrt[3]{-icx^{3}+3icwz-1}\text{, }c\neq 0$$

Solve for c

$$\left\{\begin{matrix}c=\frac{\sqrt{4a^{3}-9\left(wz\right)^{2}+6wzx^{3}-x^{6}}+ix^{3}-3iwz}{2a^{3}}\text{; }c=\frac{-\sqrt{4a^{3}-9\left(wz\right)^{2}+6wzx^{3}-x^{6}}+ix^{3}-3iwz}{2a^{3}}\text{, }&a\neq 0\\c=\frac{1}{3iwz-ix^{3}}\text{, }&a=0\text{ and }x\neq \sqrt[3]{z}\sqrt[3]{3w}\text{ and }x\neq e^{\frac{4\pi i}{3}}\sqrt[3]{z}\sqrt[3]{3w}\text{ and }x\neq e^{\frac{2\pi i}{3}}\sqrt[3]{z}\sqrt[3]{3w}\end{matrix}\right.$$