Step by step
The question
Question: add 3/(x² − 9) + 1/(x + 3).
1 Factor the bottoms
x² − 9 is a square minus a square.
So it is (x + 3)(x − 3).
We write the bottoms as factors.
2 Find the common bottom
The bottoms are (x + 3)(x − 3) and (x + 3).
The common bottom must have every factor of both bottoms, each one once.
It is (x + 3)(x − 3).
Careful!
We can't add the tops and add the bottoms.
Fractions need the same bottom first.
3 Make the bottoms the same
The first fraction already has it.
The second fraction is missing (x − 3).
We multiply its top and bottom by (x − 3).
(x − 3)/(x − 3) is 1, so the value stays the same.
4 Add the tops
Now one fraction: (3 + (x − 3)) over (x + 3)(x − 3).
5 Open the brackets
A bracket with a plus or nothing in front can just be removed.
Now we have 3 + x − 3.
6 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
Plain numbers go together.
Each colour is one kind.
We move like terms next to each other.
Each term takes its sign with it.
3 − 3 is 0.
So the expression is x.
7 Cross out the same
Is a whole factor on top the same as one on the bottom?
No.
So nothing crosses out.
We leave the bottom as factors.
So the answer is x/(x + 3)(x − 3).
The bottoms can't be 0.
So x can't be −3 or 3.
8 Check
Let's check with a number: x = 1.
First in the question.
3 over −8 plus 1/4 is −1/8.
Now in our answer.
We get −1/8 too.
Both are −1/8.
So x/(x + 3)(x − 3) is right!
Let's look back.
First: factor the bottoms.
Then: find the common bottom.
Then: make the bottoms the same.
Then: add the tops.
Then: open the brackets.
Then: group like terms.
Then: cross out the same.
Last: check.