Solve (xy/y+x-x)(x+xy/x-y):x^2+y^2/x^2-y^2

$$( \frac { x y } { y + x } - x ) ( x + \frac { x y } { x - y } ) : \frac { x ^ { 2 } + y ^ { 2 } } { x ^ { 2 } - y ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{\frac{-x^{2}}{y+x}\times \frac{x^{2}}{x-y}}{\frac{x^{2}+y^{2}}{x^{2}-y^{2}}}$$

Multiply \frac{-x^{2}}{y+x} times \frac{x^{2}}{x-y} by multiplying numerator times numerator and denominator times denominator.

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

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

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{\frac{-x^{2}}{y+x}\times \frac{x^{2}}{x-y}}{\frac{x^{2}+y^{2}}{x^{2}-y^{2}}}$$

Multiply \frac{-x^{2}}{y+x} times \frac{x^{2}}{x-y} by multiplying numerator times numerator and denominator times denominator.

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

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

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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