Question

In the following exercise, perform the indicated operation:

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

Short answer

$\frac{2q\left(q-8\right)}{\left(q+5\right)\left(q-3\right)}$

Step-by-step solution

Step 1. Given information.

The following expression is given:

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

Step 2. Perform the indicated operation.

Factorise the denominators:

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

Find out the LCD:

$\left(q+5\right)\left(q-3\right)$

Rewrite the expression with the denominators equal to the LCD:

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

Simplify:

$$\begin{gathered} \frac{2q^2-6q+3q+15-13q-15}{\left(q+5\right)\left(q-3\right)} \\ \frac{2q^2-16q}{\left(q+5\right)\left(q-3\right)} \\ \frac{2q\left(q-8\right)}{\left(q+5\right)\left(q-3\right)} \end{gathered}$$