Solve x^2+y^2/x+y-x^2-y^2/x-y

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-y^{2}}{x-y}.

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

Cancel out x-y in both numerator and denominator.

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

To find the opposite of x+y, find the opposite of each term.

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-y^{2}}{x-y}.

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

Cancel out x-y in both numerator and denominator.

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

To find the opposite of x+y, find the opposite of each term.

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

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

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

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

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

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

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

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

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