Solve 5t^2+4=sqrt{x^3+θ}

$$5 t ^ { 2 } + 4 = \sqrt { x ^ { 3 } + \theta }$$

Solve for t (complex solution)

$$t=-\frac{\sqrt{5\left(\sqrt{x^{3}+\theta }-4\right)}}{5}$$

$$t=\frac{\sqrt{5\left(\sqrt{x^{3}+\theta }-4\right)}}{5}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=e^{\frac{2\pi i}{3}}\sqrt[3]{25t^{4}+40t^{2}-\theta +16}\text{; }x=\sqrt[3]{25t^{4}+40t^{2}-\theta +16}\text{; }x=e^{\frac{4\pi i}{3}}\sqrt[3]{25t^{4}+40t^{2}-\theta +16}\text{, }&arg(5t^{2}+4)<\pi \\x=-\sqrt[3]{\theta }\text{; }x=e^{\frac{\pi i}{3}}\sqrt[3]{\theta }\text{; }x=e^{\frac{5\pi i}{3}}\sqrt[3]{\theta }\text{, }&t=-\frac{2\sqrt{5}i}{5}\text{ or }t=\frac{2\sqrt{5}i}{5}\end{matrix}\right.$$

Solve for t

$$t=\frac{\sqrt{5\left(\sqrt{x^{3}+\theta }-4\right)}}{5}$$

$$t=-\frac{\sqrt{5\left(\sqrt{x^{3}+\theta }-4\right)}}{5}\text{, }\theta \geq 16-x^{3}\text{ and }x\geq -\sqrt[3]{\theta }$$

Solve for x

$$x=\sqrt[3]{25t^{4}+40t^{2}-\theta +16}$$