Question

Simplify$\frac{2q}{q+5}+\frac{3}{q-3}-\frac{13q+15}{q^2+2q-15}$.

Short answer

The result is$\frac{2q^2-16q}{(q+5)(q-3)}$.

Step-by-step solution

Given Information      

The Given expression is$\frac{2q}{q+5}+\frac{3}{q-3}-\frac{13q+15}{q^2+2q-15}$.

Find the LCD of denominators.   

Factor the denominators.

$q^2+2q-15=(q+5)(q-3)$

Rewrite the expression in factored form.

$=\frac{2q}{q+5}+\frac{3}{q-3}-\frac{13q+15}{(q+5)(q-3)}$

Rewrite each fraction as an equivalent fraction with the LCD.  

$$\begin{gathered} =\frac{2q(q-3)}{(q+5)(q-3)}+\frac{3(q+5)}{(q+5)(q-3)}-\frac{13q+15}{(q+5)(q-3)} \\ =\frac{2q^2-6q}{(q+5)(q-3)}+\frac{3q+15}{(q+5)(q-3)}-\frac{13q+15}{(q+5)(q-3)} \end{gathered}$$

Simplify. 

Take common denominator.

$$\begin{gathered} =\frac{2q^2-6q+3q+15-(13q+15)}{(q+5)(q-3)} \\ =\frac{2q^2-3q+15-13q-15}{(q+5)(q-3)} \\ =\frac{2q^2-16q}{(q+5)(q-3)} \end{gathered}$$