Solve y^2=x+y^4

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

Solve for x

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

Solution Steps

Swap sides so that all variable terms are on the left hand side.

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

Subtract y^{4} from both sides.

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

Solve for y (complex solution)

$$y=\frac{\sqrt{2\sqrt{1-4x}+2}}{2}$$

$$y=-\frac{\sqrt{2\sqrt{1-4x}+2}}{2}$$

$$y=-\frac{\sqrt{-2\sqrt{1-4x}+2}}{2}$$

$$y=\frac{\sqrt{-2\sqrt{1-4x}+2}}{2}$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{\sqrt{-2\sqrt{1-4x}+2}}{2}\text{; }y=\frac{\sqrt{-2\sqrt{1-4x}+2}}{2}\text{, }&x\geq 0\text{ and }x\leq \frac{1}{4}\\y=-\frac{\sqrt{2\sqrt{1-4x}+2}}{2}\text{; }y=\frac{\sqrt{2\sqrt{1-4x}+2}}{2}\text{, }&x\leq \frac{1}{4}\end{matrix}\right.$$