Solve y=x^4-2mx^2+2m

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

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{x^{4}-y}{2\left(1-x^{2}\right)}\text{, }&x\neq 1\text{ and }x\neq -1\\m\in \mathrm{C}\text{, }&\left(x=-1\text{ or }x=1\right)\text{ and }y=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{4}-2mx^{2}+2m=y$$

Subtract x^{4} from both sides.

$$-2mx^{2}+2m=y-x^{4}$$

Combine all terms containing m.

$$\left(-2x^{2}+2\right)m=y-x^{4}$$

The equation is in standard form.

$$\left(2-2x^{2}\right)m=y-x^{4}$$

Divide both sides by -2x^{2}+2.

$$\frac{\left(2-2x^{2}\right)m}{2-2x^{2}}=\frac{y-x^{4}}{2-2x^{2}}$$

Dividing by -2x^{2}+2 undoes the multiplication by -2x^{2}+2.

$$m=\frac{y-x^{4}}{2-2x^{2}}$$

Divide y-x^{4} by -2x^{2}+2.

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

Solve for m

$$\left\{\begin{matrix}m=-\frac{x^{4}-y}{2\left(1-x^{2}\right)}\text{, }&|x|\neq 1\\m\in \mathrm{R}\text{, }&y=1\text{ and }|x|=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{4}-2mx^{2}+2m=y$$

Subtract x^{4} from both sides.

$$-2mx^{2}+2m=y-x^{4}$$

Combine all terms containing m.

$$\left(-2x^{2}+2\right)m=y-x^{4}$$

The equation is in standard form.

$$\left(2-2x^{2}\right)m=y-x^{4}$$

Divide both sides by -2x^{2}+2.

$$\frac{\left(2-2x^{2}\right)m}{2-2x^{2}}=\frac{y-x^{4}}{2-2x^{2}}$$

Dividing by -2x^{2}+2 undoes the multiplication by -2x^{2}+2.

$$m=\frac{y-x^{4}}{2-2x^{2}}$$

Divide y-x^{4} by -2x^{2}+2.

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

Solve for x (complex solution)

$$x=\sqrt{-\sqrt{y+m^{2}-2m}+m}$$

$$x=-\sqrt{-\sqrt{y+m^{2}-2m}+m}$$

$$x=-\sqrt{\sqrt{y+m^{2}-2m}+m}$$

$$x=\sqrt{\sqrt{y+m^{2}-2m}+m}$$

Solve for x

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