Solve x-2y/x^2+2xy-x+2y/x^2-2xy/frac{4y^2{4y^2-{x}^2}}

$$\frac{ \frac{ x-2y }{ { x }^{ 2 } +2xy } - \frac{ x+2y }{ { x }^{ 2 } -2xy } }{ \frac{ 4 { y }^{ 2 } }{ 4 { y }^{ 2 } - { x }^{ 2 } } }$$

Evaluate

$$\frac{2}{y}$$

Short Solution Steps

Factor x^{2}+2xy. Factor x^{2}-2xy.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2y\right) and x\left(x-2y\right) is x\left(x-2y\right)\left(x+2y\right). Multiply \frac{x-2y}{x\left(x+2y\right)} times \frac{x-2y}{x-2y}. Multiply \frac{x+2y}{x\left(x-2y\right)} times \frac{x+2y}{x+2y}.

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

Since \frac{\left(x-2y\right)\left(x-2y\right)}{x\left(x-2y\right)\left(x+2y\right)} and \frac{\left(x+2y\right)\left(x+2y\right)}{x\left(x-2y\right)\left(x+2y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2y\right)\left(x-2y\right)-\left(x+2y\right)\left(x+2y\right).

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

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

$$\frac{\frac{-8xy}{x\left(x-2y\right)\left(x+2y\right)}}{\frac{4y^{2}}{4y^{2}-x^{2}}}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{-8y}{\left(x-2y\right)\left(x+2y\right)}}{\frac{4y^{2}}{4y^{2}-x^{2}}}$$

Divide \frac{-8y}{\left(x-2y\right)\left(x+2y\right)} by \frac{4y^{2}}{4y^{2}-x^{2}} by multiplying \frac{-8y}{\left(x-2y\right)\left(x+2y\right)} by the reciprocal of \frac{4y^{2}}{4y^{2}-x^{2}}.

$$\frac{-8y\left(4y^{2}-x^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4y^{2}}$$

Cancel out 4y in both numerator and denominator.

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

Factor the expressions that are not already factored.

$$\frac{-2\left(x-2y\right)\left(-x-2y\right)}{y\left(x-2y\right)\left(x+2y\right)}$$

Extract the negative sign in -x-2y.

$$\frac{-2\left(-1\right)\left(x-2y\right)\left(x+2y\right)}{y\left(x-2y\right)\left(x+2y\right)}$$

Cancel out \left(x-2y\right)\left(x+2y\right) in both numerator and denominator.

$$\frac{-2\left(-1\right)}{y}$$

Expand the expression.

$$\frac{2}{y}$$

Expand

$$\frac{2}{y}$$

Short Solution Steps

Factor x^{2}+2xy. Factor x^{2}-2xy.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2y\right) and x\left(x-2y\right) is x\left(x-2y\right)\left(x+2y\right). Multiply \frac{x-2y}{x\left(x+2y\right)} times \frac{x-2y}{x-2y}. Multiply \frac{x+2y}{x\left(x-2y\right)} times \frac{x+2y}{x+2y}.

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

Since \frac{\left(x-2y\right)\left(x-2y\right)}{x\left(x-2y\right)\left(x+2y\right)} and \frac{\left(x+2y\right)\left(x+2y\right)}{x\left(x-2y\right)\left(x+2y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-2y\right)\left(x-2y\right)-\left(x+2y\right)\left(x+2y\right).

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

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

$$\frac{\frac{-8xy}{x\left(x-2y\right)\left(x+2y\right)}}{\frac{4y^{2}}{4y^{2}-x^{2}}}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{-8y}{\left(x-2y\right)\left(x+2y\right)}}{\frac{4y^{2}}{4y^{2}-x^{2}}}$$

Divide \frac{-8y}{\left(x-2y\right)\left(x+2y\right)} by \frac{4y^{2}}{4y^{2}-x^{2}} by multiplying \frac{-8y}{\left(x-2y\right)\left(x+2y\right)} by the reciprocal of \frac{4y^{2}}{4y^{2}-x^{2}}.

$$\frac{-8y\left(4y^{2}-x^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4y^{2}}$$

Cancel out 4y in both numerator and denominator.

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

Factor the expressions that are not already factored.

$$\frac{-2\left(x-2y\right)\left(-x-2y\right)}{y\left(x-2y\right)\left(x+2y\right)}$$

Extract the negative sign in -x-2y.

$$\frac{-2\left(-1\right)\left(x-2y\right)\left(x+2y\right)}{y\left(x-2y\right)\left(x+2y\right)}$$

Cancel out \left(x-2y\right)\left(x+2y\right) in both numerator and denominator.

$$\frac{-2\left(-1\right)}{y}$$

Expand the expression.

$$\frac{2}{y}$$