Step by step
The question
Question: simplify (x² − x − 6)/(x² − 9).
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.
We do the same with brackets.
1 Factor the top
We look at the top: x² − x − 6.
For x² − 1x − 6, we need two numbers.
They multiply to −6, and they add to −1.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 1.
2 and 3 give 1.
Yes!
With the signs: 2 and −3.
So the top is (x + 2)(x − 3).
2 Factor the bottom
We look at the bottom: x² − 9.
x² − 9 is x² minus 3 squared.
A square minus a square is the plus bracket times the minus bracket.
So the bottom is (x + 3)(x − 3).
3 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.
Now we write the fraction with the factors: (x + 2)(x − 3)/(x + 3)(x − 3).
(x − 3)/(x − 3) is 1.
Cross them out.
4 Write what is left
What is left is (x + 2)/(x + 3).
One more thing: the bottom of the question can't be 0.
So x can't be −3 or 3.
5 Check
Let's check with a number: x = 2.
First in the question.
We get 4/5.
Now in our answer.
We get 4/5 too.
Both are 4/5.
So (x + 2)/(x + 3) is right!
Let's look back.
First: factor the top.
Then: factor the bottom.
Then: cross out the same.
Then: write what is left.
Last: check.