Solve 1/y=x^2+1/x^2

$$\frac { 1 } { y } = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }$$

Solve for y (complex solution)

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

$$x\neq 0\text{ and }x\neq \sqrt{2}\left(-\frac{1}{2}-\frac{1}{2}i\right)\text{ and }x\neq \sqrt{2}\left(\frac{1}{2}+\frac{1}{2}i\right)\text{ and }x\neq \sqrt{2}\left(-\frac{1}{2}+\frac{1}{2}i\right)\text{ and }x\neq \sqrt{2}\left(\frac{1}{2}-\frac{1}{2}i\right)$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by yx^{2}, the least common multiple of y,x^{2}.

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

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

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

Combine all terms containing y.

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

Divide both sides by x^{4}+1.

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

Dividing by x^{4}+1 undoes the multiplication by x^{4}+1.

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

Variable y cannot be equal to 0.

$$y=\frac{x^{2}}{x^{4}+1}\text{, }y\neq 0$$

Solve for y

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

$$x\neq 0$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by yx^{2}, the least common multiple of y,x^{2}.

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

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

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

Combine all terms containing y.

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

Divide both sides by x^{4}+1.

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

Dividing by x^{4}+1 undoes the multiplication by x^{4}+1.

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

Variable y cannot be equal to 0.

$$y=\frac{x^{2}}{x^{4}+1}\text{, }y\neq 0$$

Solve for x (complex solution)

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

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

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

$$x=\frac{iy^{-\frac{1}{2}}\sqrt{2\left(\sqrt{1-4y^{2}}-1\right)}}{2}\text{, }y\neq 0$$

Solve for x

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

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

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

$$x=-\frac{\sqrt{\frac{2\left(-\sqrt{1-4y^{2}}+1\right)}{y}}}{2}\text{, }y>0\text{ and }y\leq \frac{1}{2}$$