Question

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

Answer 1

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

`- (5x-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:`9/(x^2-9) - 5/(x+3)` .............(1)

Consider `x^2-9` ....................(2)

But `9 = 3^2` .....................(3)

Substitute (3) into (2) giving:

`x^2 - 3^3`

But `(x^2-3^2)=(x-3)(x+3)`........(4)

Substitute (4) into (1) giving:

`9/((x-3)(x+3)) - 5/(x+3)`

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

`(9-5(x-3))/((x+3)(x-3))`

`(9-5x+15)/((x+3)(x-3))`

`((-5x)+24)/ ((x+3)(x-3))` .................(5)

but `(-5x+24) = -(5x-24)`...............(6)

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

`- (5x-24)/(x^2-9)`

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