Solve x=frac{2sqrt{3}}{3}sqrt{2t^2-t-1/t^2}=frac{2sqrt{3}}{3}sqrt{-{left(1/tright)}^2-1/t+2}

$$x = \frac{ 2 \sqrt{ 3 } }{ 3 } \sqrt{ \frac{ 2 { t }^{ 2 } -t-1 }{ { t }^{ 2 } } } = \frac{ 2 \sqrt{ 3 } }{ 3 } \sqrt{ - { \left( \frac{ 1 }{ t } \right) }^{ 2 } - \frac{ 1 }{ t } +2 }$$

Evaluate (complex solution)

$$x=\frac{2\sqrt{6-\frac{3}{t}-\frac{3}{t^{2}}}}{3}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{2\sqrt{6t^{2}-3t-3}}{3t}\text{, }&t\geq 1\\x=-\frac{2\sqrt{6t^{2}-3t-3}}{3t}\text{, }&t\leq -\frac{1}{2}\end{matrix}\right.$$

Solve for t

$$\left\{\begin{matrix}t=-1\text{, }&x=\frac{2\sqrt{6}}{3}\\t=\frac{2\left(-\sqrt{3\left(3-x^{2}\right)}+1\right)}{8-3x^{2}}\text{; }t=\frac{2\left(\sqrt{3\left(3-x^{2}\right)}+1\right)}{8-3x^{2}}\text{, }&x\leq \sqrt{3}\text{ and }x\geq 0\text{ and }x\neq \frac{2\sqrt{6}}{3}\end{matrix}\right.$$