Solve x/x-2y+x/x+2y+4xy/4y^2-x^2

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Factor 4y^{2}-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2y\right)\left(x+2y\right) and \left(x+2y\right)\left(-x+2y\right) is \left(x+2y\right)\left(-x+2y\right). Multiply \frac{2x^{2}}{\left(x-2y\right)\left(x+2y\right)} times \frac{-1}{-1}.

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

Since \frac{-2x^{2}}{\left(x+2y\right)\left(-x+2y\right)} and \frac{4xy}{\left(x+2y\right)\left(-x+2y\right)} have the same denominator, add them by adding their numerators.

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

Factor the expressions that are not already factored in \frac{-2x^{2}+4xy}{\left(x+2y\right)\left(-x+2y\right)}.

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

Cancel out -x+2y in both numerator and denominator.

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

Factor

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