Solve sqrt{6/x-8}=a^2-8z

$$\sqrt{ \frac{ 6 }{ x-8 } } = { a }^{ 2 } -8z$$

Solve for a

$$a=\frac{\sqrt{8\sqrt{x-8}z+\sqrt{6}}}{\sqrt[4]{x-8}}$$

$$a=-\frac{\sqrt{8\sqrt{x-8}z+\sqrt{6}}}{\sqrt[4]{x-8}}\text{, }\left(z=0\text{ and }x>8\right)\text{ or }\left(x>8\text{ and }x\leq \frac{3\left(\frac{256z^{2}}{3}+1\right)}{32z^{2}}\right)\text{ or }\left(x=\frac{3\left(\frac{256z^{2}}{3}+1\right)}{32z^{2}}\text{ and }z\neq 0\right)$$

Solve for x

$$x=-\frac{2\left(-256z^{2}+64za^{2}-4a^{4}-3\right)}{\left(8z-a^{2}\right)^{2}}$$

$$z\neq \frac{a^{2}}{8}\text{ and }-\left(8z-a^{2}\right)\geq 0$$