Solve a/a+2-a/a-2-2a^2/4-a^2

$$\frac { a } { a + 2 } - \frac { a } { a - 2 } - \frac { 2 a ^ { 2 } } { 4 - a ^ { 2 } }$$

Evaluate

$$\frac{2a}{a+2}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in a^{2}-2a-a^{2}-2a.

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

Factor 4-a^{2}.

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

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

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

Since \frac{-4a}{\left(a-2\right)\left(a+2\right)} and \frac{-2a^{2}}{\left(a-2\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -4a-\left(-2a^{2}\right).

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

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

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

Cancel out a-2 in both numerator and denominator.

$$\frac{2a}{a+2}$$

Factor

$$\frac{2a}{a+2}$$