Solve {left(6/4yright)}^2+{left(y-6/4sqrt{sqrt{x^2}}right)}^2=1

$${ \left( \frac{ 6 }{ 4 } y \right) }^{ 2 } + { \left(y- \frac{ 6 }{ 4 } \sqrt{ \sqrt{ { x }^{ 2 } } } \right) }^{ 2 } =1$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=0\text{, }&y=-\frac{2\sqrt{13}}{13}\text{ or }y=\frac{2\sqrt{13}}{13}\\x=-\frac{\left(-\sqrt{4-9y^{2}}+2y\right)^{2}}{9}\text{; }x=\frac{\left(-\sqrt{4-9y^{2}}+2y\right)^{2}}{9}\text{, }&arg(\frac{-\sqrt{4-9y^{2}}+2y}{3})<\frac{\pi }{2}\\x=-\frac{\left(\sqrt{4-9y^{2}}+2y\right)^{2}}{9}\text{; }x=\frac{\left(\sqrt{4-9y^{2}}+2y\right)^{2}}{9}\text{, }&arg(\frac{\sqrt{4-9y^{2}}+2y}{3})<\frac{\pi }{2}\end{matrix}\right.$$

Solve for y (complex solution)

$$y=\frac{\sqrt{-81\sqrt{x^{2}}+52}+6\sqrt[4]{x^{2}}}{13}$$

$$y=\frac{-\sqrt{-81\sqrt{x^{2}}+52}+6\sqrt[4]{x^{2}}}{13}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\left(-\sqrt{4-9y^{2}}+2y\right)^{2}}{9}\text{; }x=-\frac{\left(-\sqrt{4-9y^{2}}+2y\right)^{2}}{9}\text{, }&\frac{-\sqrt{4-9y^{2}}+2y}{3}\geq 0\text{ and }|y|\leq \frac{2}{3}\\x=\frac{\left(\sqrt{4-9y^{2}}+2y\right)^{2}}{9}\text{; }x=-\frac{\left(\sqrt{4-9y^{2}}+2y\right)^{2}}{9}\text{, }&\frac{\sqrt{4-9y^{2}}+2y}{3}\geq 0\text{ and }|y|\leq \frac{2}{3}\end{matrix}\right.$$

Solve for y

$$y=\frac{\sqrt{-81|x|+52}+6\sqrt{|x|}}{13}$$

$$y=\frac{-\sqrt{-81|x|+52}+6\sqrt{|x|}}{13}\text{, }|x|\leq \frac{52}{81}$$