Solve (3a/a+2-a/a-2)-2a/a^2-4

$$( \frac { 3 a } { a + 2 } - \frac { a } { a - 2 } ) - \frac { 2 a } { a ^ { 2 } - 4 }$$

Evaluate

$$\frac{2a\left(a-5\right)}{a^{2}-4}$$

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{3a}{a+2} times \frac{a-2}{a-2}. Multiply \frac{a}{a-2} times \frac{a+2}{a+2}.

$$\frac{3a\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{a\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{a^{2}-4}$$

Since \frac{3a\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.

$$\frac{3a\left(a-2\right)-a\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{a^{2}-4}$$

Do the multiplications in 3a\left(a-2\right)-a\left(a+2\right).

$$\frac{3a^{2}-6a-a^{2}-2a}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{a^{2}-4}$$

Combine like terms in 3a^{2}-6a-a^{2}-2a.

$$\frac{2a^{2}-8a}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{a^{2}-4}$$

Factor a^{2}-4.

$$\frac{2a^{2}-8a}{\left(a-2\right)\left(a+2\right)}-\frac{2a}{\left(a-2\right)\left(a+2\right)}$$

Since \frac{2a^{2}-8a}{\left(a-2\right)\left(a+2\right)} and \frac{2a}{\left(a-2\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2a^{2}-8a-2a}{\left(a-2\right)\left(a+2\right)}$$

Combine like terms in 2a^{2}-8a-2a.

$$\frac{2a^{2}-10a}{\left(a-2\right)\left(a+2\right)}$$

Expand \left(a-2\right)\left(a+2\right).

$$\frac{2a^{2}-10a}{a^{2}-4}$$

Factor

$$\frac{2a\left(a-5\right)}{a^{2}-4}$$