Solve a/a-b+b/a+b-2ab/a^2-b^2

$$\frac { a } { a - b } + \frac { b } { a + b } - \frac { 2 a b } { a ^ { 2 } - b ^ { 2 } }$$

Evaluate

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Short Solution Steps

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

$$\frac{a\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}+\frac{b\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

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

$$\frac{a\left(a+b\right)+b\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Do the multiplications in a\left(a+b\right)+b\left(a-b\right).

$$\frac{a^{2}+ab+ab-b^{2}}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Combine like terms in a^{2}+ab+ab-b^{2}.

$$\frac{a^{2}-b^{2}+2ab}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Factor a^{2}-b^{2}.

$$\frac{a^{2}-b^{2}+2ab}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{\left(a+b\right)\left(a-b\right)}$$

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

$$\frac{a^{2}-b^{2}+2ab-2ab}{\left(a+b\right)\left(a-b\right)}$$

Combine like terms in a^{2}-b^{2}+2ab-2ab.

$$\frac{a^{2}-b^{2}}{\left(a+b\right)\left(a-b\right)}$$

Factor the expressions that are not already factored in \frac{a^{2}-b^{2}}{\left(a+b\right)\left(a-b\right)}.

$$\frac{\left(a+b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}$$

Cancel out \left(a+b\right)\left(a-b\right) in both numerator and denominator.

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Factor

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