Solve y=(r^2+11)^2/5-r^4

$$y = \frac { ( r ^ { 2 } + 11 ) ^ { 2 } } { 5 - r ^ { 4 } }$$

Solve for r (complex solution)

$$\left\{\begin{matrix}r=i\left(y+1\right)^{-\frac{1}{2}}\sqrt{-\sqrt{y\left(5y-116\right)}+11}\text{; }r=-i\left(y+1\right)^{-\frac{1}{2}}\sqrt{-\sqrt{y\left(5y-116\right)}+11}\text{; }r=-i\left(y+1\right)^{-\frac{1}{2}}\sqrt{\sqrt{y\left(5y-116\right)}+11}\text{; }r=i\left(y+1\right)^{-\frac{1}{2}}\sqrt{\sqrt{y\left(5y-116\right)}+11}\text{, }&y\neq -1\\r=-\frac{3\sqrt{77}i}{11}\approx -0-2.393172106i\text{; }r=\frac{3\sqrt{77}i}{11}\approx 2.393172106i\text{, }&y=-1\end{matrix}\right.$$

Solve for y (complex solution)

$$y=\frac{\left(r^{2}+11\right)^{2}}{5-r^{4}}$$

$$r\neq -\sqrt[4]{5}i\text{ and }r\neq -\sqrt[4]{5}\text{ and }r\neq \sqrt[4]{5}i\text{ and }r\neq \sqrt[4]{5}$$

Solve for r

$$\left\{\begin{matrix}r=-\sqrt{-\frac{\sqrt{-y\left(116-5y\right)}+11}{y+1}}\text{; }r=\sqrt{-\frac{\sqrt{-y\left(116-5y\right)}+11}{y+1}}\text{, }&y<-1\text{ and }\sqrt{-\frac{\sqrt{-y\left(116-5y\right)}+11}{y+1}}\neq \sqrt[4]{5}\\r=-\sqrt{\frac{\sqrt{y\left(5y-116\right)}-11}{y+1}}\text{; }r=\sqrt{\frac{\sqrt{y\left(5y-116\right)}-11}{y+1}}\text{, }&y\geq \frac{121}{5}\end{matrix}\right.$$

Solve for y

$$y=\frac{\left(r^{2}+11\right)^{2}}{5-r^{4}}$$

$$|r|\neq \sqrt[4]{5}$$