Question

In the following exercise, perform the indicated operation:

$\frac{c}{c+2}+\frac{5}{c-2}-\frac{10c}{c^2-4}$

Short answer

The simplified answer is$\frac{c-5}{c+2}$

Step-by-step solution

Step 1. Given information.

The following expression is given:

$\frac{c}{c+2}+\frac{5}{c-2}-\frac{10c}{c^2-4}$

Step 2. Performing the indicated operation.

Factorise the denominators:

$\frac{c}{c+2}+\frac{5}{c-2}-\frac{10c}{\left(c+2\right)\left(c-2\right)}$

Find out the LCD:

$\left(c+2\right)\left(c-2\right)$

Rewrite the expression with the denominators equal to the LCD:

role="math" localid="1644733589455" $\frac{c\left(c-2\right)}{\left(c+2\right)\left(c-2\right)}+\frac{5\left(c+2\right)}{\left(c-2\right)\left(c+2\right)}-\frac{10c}{\left(c+2\right)\left(c-2\right)}$

Simplify:

role="math" localid="1644733614175" $$\begin{gathered} \frac{c^2-2c+5c+10-10c}{\left(c+2\right)\left(c-2\right)} \\ \frac{c^2-7c+10}{\left(c+2\right)\left(c-2\right)} \\ \frac{\left(c-5\right)\left(c-2\right)}{\left(c+2\right)\left(c-2\right)} \end{gathered}$$

Cancel common factors from the numerator and the denominator:

$\frac{c-5}{c+2}$