Question

Simplify$\frac{c}{c+2}+\frac{5}{c-2}-\frac{11c}{c^2-4}$.

Short answer

The result is$\frac{c-5}{c+2}$.

Step-by-step solution

Given Information       

The Given expression is$\frac{c}{c+2}+\frac{5}{c-2}-\frac{11c}{c^2-4}$.

Find the LCD of denominators.   

Factor the denominators.

$c^2-4=(c+2)(c-2)$

LCD$=(c+2)(c-2)$

Rewrite the expression in factored form.

$=\frac{c}{c+2}+\frac{5}{c-2}-\frac{11c}{(c+2)(c-2)}$

Rewrite each fraction as an equivalent fraction with the LCD.   

$$\begin{gathered} =\frac{c(c-2)}{(c+2)(c-2)}+\frac{5(c+2)}{(c+2)(c-2)}-\frac{11c}{(c+2)(c-2)} \\ =\frac{c^2-2c}{(c+2)(c-2)}+\frac{5c+10}{(c+2)(c-2)}-\frac{11c}{(c+2)(c-2)} \end{gathered}$$

Simplify.  

Take common denominator.

$$\begin{gathered} =\frac{c^2-2c+5c+10-11c}{(c+2)(c-2)} \\ =\frac{c^2-8c+10}{(c+2)(c-2)} \end{gathered}$$

Factor numerator and cancel common factors.

$$\begin{gathered} =\frac{(c-5)(c-2)}{(c+2)(c-2)} \\ =\frac{c-5}{c+2} \end{gathered}$$