Question

In the following exercises, perform the indicated operations

$\frac{3x}{x^2-9}+\frac{5}{x^2+6x+9}$

Short answer

The simplified expression is$\frac{3x^2+14x-15}{{(x+3)}^2(x-3)}$

Step-by-step solution

Step 1. Given information

Expression :

$\frac{3x}{x^2-9}+\frac{5}{x^2+6x+9}$

Step 2. Find LCD  

Denominator of first fraction is$x^2-9$

Factorize this denominator

$x^2-3^2=(x-3)(x+3)$

Denominator of the second fraction is$x^2+6x+9$

Factorize this denominator

$x^2+2.3.x+3^2={(x+3)}^2$

here $\left(x+3\right)$is common between both denominator

So the LCD is${\left(x+3\right)}^2\left(x-3\right)$

Step 3. Rewrite each rational expression as  an equivalent rational expression with the LCD   

$\frac{3x(x+3)}{{(x+3)}^2(x-3)}+\frac{5(x-3)}{{(x+3)}^2(x-3)}$

Step 4.Simplify the numerators.   

$\frac{3x^2+9x}{{(x+3)}^2(x-3)}+\frac{5x-15}{{(x+3)}^2(x-3)}$

Step 5. Add numerators of the fractions   

$\frac{3x^2+9x+5x-15}{{(x+3)}^2(x-3)}$

Step 6. Simplify numerator 

$\frac{3x^2+14x-15}{{(x+3)}^2(x-3)}$