Step by step
The question
Question: simplify (x² − 9)/(x + 3).
A fraction with x in it is called a rational expression.
We make it simpler like a number fraction.
6/9: both have a 3 in them.
3/3 is 1, so 2/3 is left.
Next we do the same with x and brackets.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Factor the top
We look at the top: x² − 9.
Factor means: write it as a times, with brackets.
x² − 9: 9 is 3 times 3, so it is x² minus 3 squared.
A square minus a square is the plus bracket times the minus bracket.
Why?
We open the brackets to see.
−3x and +3x make zero.
Only x² − 9 is left.
So the top is (x + 3)(x − 3).
2 Cross out the same
Careful!
We cross out only a whole bracket, or a whole x, that multiplies everything.
Never one part of a bracket.
A number check: 2 plus 3, all over 3, is 5/3.
Crossing out the 3s would give 2.
Wrong!
Now we write the fraction with the factors: (x + 3)(x − 3)/(x + 3).
(x + 3)/(x + 3) is 1.
Cross them out.
3 Write what is left
What is left is x − 3.
One more thing: the bottom of the question can't be 0: we can't divide by 0.
x + 3 is 0 when x is −3.
These come from the question, so they stay.
So x can't be −3.
4 Check
Let's check with a number: x = 2.
We pick a small number that makes no bottom 0.
First in the question.
We get −1.
Now in our answer.
We get −1 too.
Both are −1.
So x − 3 is right!
5 Pick the choice
Our answer is x − 3.
That is choice B.
6 Why not the others
A is x + 3.
Wrong: (x + 3)(x − 3)/(x + 3) leaves x − 3.
C is x² − 3.
Wrong: 9 ÷ 3: parts of a sum can't be cancelled.
D is x − 9.
Wrong: x² ÷ x: parts of a sum can't be cancelled.
Let's look back.
First: factor the top.
Then: cross out the same.
Then: write what is left.
Then: check.
Then: pick the choice.
Last: why not the others.