# How do you subtract 9/(x^2-9) - 5/ (x+3)?

Oct 29, 2015

I have shown every step in 'over the top' detail so you can see where everything comes from.

$- \frac{5 x - 24}{{x}^{2} - 9}$

#### Explanation:

When presented with questions you look for links which means you have to build up a 'toolbox' of remembered facts and techniques:

Given:$\frac{9}{{x}^{2} - 9} - \frac{5}{x + 3}$ .............(1)

Consider ${x}^{2} - 9$ ....................(2)

But $9 = {3}^{2}$ .....................(3)

Substitute (3) into (2) giving:

${x}^{2} - {3}^{3}$

But $\left({x}^{2} - {3}^{2}\right) = \left(x - 3\right) \left(x + 3\right)$........(4)

Substitute (4) into (1) giving:

$\frac{9}{\left(x - 3\right) \left(x + 3\right)} - \frac{5}{x + 3}$

Now both denominators have something in common so we can combine them

$\frac{9 - 5 \left(x - 3\right)}{\left(x + 3\right) \left(x - 3\right)}$

$\frac{9 - 5 x + 15}{\left(x + 3\right) \left(x - 3\right)}$

$\frac{\left(- 5 x\right) + 24}{\left(x + 3\right) \left(x - 3\right)}$ .................(5)

but $\left(- 5 x + 24\right) = - \left(5 x - 24\right)$...............(6)

substituting (6) into (5) and rewriting the expression gives:

$- \frac{5 x - 24}{{x}^{2} - 9}$