Solve (y-1/y)^2+2=y^2+1/y^2

$$( y - 1 / y ) ^ { 2 } + 2 = y ^ { 2 } + 1 / y ^ { 2 }$$

Solve for y

$$y\neq 0$$

Short Solution Steps

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{y}{y}.

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

Since \frac{yy}{y} and \frac{1}{y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in yy-1.

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

To raise \frac{y^{2}-1}{y} to a power, raise both numerator and denominator to the power and then divide.

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

Express y^{2}\times \frac{\left(y^{2}-1\right)^{2}}{y^{2}} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{2}\times 2 times \frac{y^{2}}{y^{2}}.

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

Since \frac{y^{2}\left(y^{2}-1\right)^{2}}{y^{2}} and \frac{y^{2}\times 2y^{2}}{y^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in y^{2}\left(y^{2}-1\right)^{2}+y^{2}\times 2y^{2}.

$$\frac{y^{6}-2y^{4}+y^{2}+2y^{4}}{y^{2}}=y^{4}+1$$

Combine like terms in y^{6}-2y^{4}+y^{2}+2y^{4}.

$$\frac{y^{6}+y^{2}}{y^{2}}=y^{4}+1$$

Subtract y^{4} from both sides.

$$\frac{y^{6}+y^{2}}{y^{2}}-y^{4}=1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y^{4} times \frac{y^{2}}{y^{2}}.

$$\frac{y^{6}+y^{2}}{y^{2}}-\frac{y^{4}y^{2}}{y^{2}}=1$$

Since \frac{y^{6}+y^{2}}{y^{2}} and \frac{y^{4}y^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{6}+y^{2}-y^{4}y^{2}}{y^{2}}=1$$

Do the multiplications in y^{6}+y^{2}-y^{4}y^{2}.

$$\frac{y^{6}+y^{2}-y^{6}}{y^{2}}=1$$

Combine like terms in y^{6}+y^{2}-y^{6}.

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

Subtract 1 from both sides.

$$\frac{y^{2}}{y^{2}}-1=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

$$\frac{y^{2}}{y^{2}}-\frac{y^{2}}{y^{2}}=0$$

Since \frac{y^{2}}{y^{2}} and \frac{y^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{2}-y^{2}}{y^{2}}=0$$

Combine like terms in y^{2}-y^{2}.

$$\frac{0}{y^{2}}=0$$

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

$$0=0$$

This is true for any y.

$$y\in \mathrm{R}$$

Variable y cannot be equal to 0.

$$y\in \mathrm{R}\setminus 0$$