Question

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

Short answer

The result is$\frac{4b+1}{2(b-5)}$.

Step-by-step solution

Given Information   

The Given expression is$\frac{2b}{b-5}+\frac{3}{2b}-\frac{2b-15}{2b^2-10b}$.

Find the LCD of denominators.    

Factor the denominators.

$2b^2-10b=2b(b-5)$

LCD$=2b(b-5)$

Rewrite the expression in factored form.

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

Rewrite each fraction as an equivalent fraction with the LCD.   

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

Simplify.  

Take common denominator.

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

Cancel common factors.

$$\begin{gathered} =\frac{4b^2+b}{2b(b-5)} \\ =\frac{b(4b+1)}{2b(b-5)} \\ =\frac{4b+1}{2(b-5)} \end{gathered}$$