Solve y=x^2+a^4x^-2

$$y = x ^ { 2 } + a ^ { 4 } x ^ { - 2 }$$

Solve for a (complex solution)

$$a=i\sqrt[4]{x^{2}\left(y-x^{2}\right)}$$

$$a=\sqrt[4]{x^{2}\left(y-x^{2}\right)}$$

$$a=-\sqrt[4]{x^{2}\left(y-x^{2}\right)}$$

$$a=-i\sqrt[4]{x^{2}\left(y-x^{2}\right)}\text{, }x\neq 0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{2\left(\sqrt{y^{2}-4a^{4}}+y\right)}}{2}\text{; }x=\frac{\sqrt{2\left(\sqrt{y^{2}-4a^{4}}+y\right)}}{2}\text{, }&\left(arg(y)<\pi \text{ and }y\neq 0\right)\text{ or }a\neq 0\\x=-\frac{\sqrt{2\left(-\sqrt{y^{2}-4a^{4}}+y\right)}}{2}\text{; }x=\frac{\sqrt{2\left(-\sqrt{y^{2}-4a^{4}}+y\right)}}{2}\text{, }&\left(|\frac{arg(y^{2})}{2}-arg(y)|\geq \pi \text{ and }y\neq 0\right)\text{ or }a\neq 0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=\sqrt{x\sqrt{y-x^{2}}}\text{; }a=-\sqrt{x\sqrt{y-x^{2}}}\text{, }&\left(y=x^{2}\text{ and }x\neq 0\right)\text{ or }\left(y\geq x^{2}\text{ and }x>0\right)\\a=\sqrt{-x\sqrt{y-x^{2}}}\text{; }a=-\sqrt{-x\sqrt{y-x^{2}}}\text{, }&\left(y=x^{2}\text{ and }x\neq 0\right)\text{ or }\left(y\geq x^{2}\text{ and }x<0\right)\end{matrix}\right.$$

Solve for x

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