Question

In the following exercise, perform the indicated operation:

$\frac{2b}{b-5}+\frac{3}{2b}-\frac{2b-15}{2b^2-10b}$

Short answer

$\frac{4b+1}{2\left(b+5\right)}$

Step-by-step solution

Step 1. Given information.

The following expression is given:

$\frac{2b}{b-5}+\frac{3}{2b}-\frac{2b-15}{2b^2-10b}$

Step 2. Performing the given operation.

Factorise the denominator:

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

Find out the LCD of the denominators:

$2b\left(b-5\right)$

Rewrite the expression with denominators equal to the LCD:

$\frac{2b.2b}{2b\left(b-5\right)}+\frac{3\left(b-5\right)}{2b\left(b-5\right)}-\frac{2b-15}{2b\left(b-5\right)}$

Simplify:

$$\begin{gathered} \frac{4b^2+3b-15-2b+15}{2b\left(b-5\right)} \\ \frac{4b^2+b}{2b\left(b-5\right)} \\ \frac{b\left(4b+1\right)}{2b\left(b-5\right)} \end{gathered}$$

Cancel the common factors from the numerator and the denominator:

$\frac{(4b+1)}{2(b-5)}$