Step by step
The question
Question: simplify (2x² − 18)/(x² + 6x + 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: 2x² − 18.
2 and 18 can both be divided by 2.
We take 2 out.
What is left goes in the bracket.
x² − 9 is x² minus 3 squared.
A square minus a square is the plus bracket times the minus bracket.
So the top is 2(x + 3)(x − 3).
2 Factor the bottom
We look at the bottom: x² + 6x + 9.
For x² + 6x + 9, we need two numbers.
They multiply to 9, and they add to 6.
They multiply to a plus number, so the two numbers have the same sign.
They add to a plus number, so both are plus.
We list the pairs that make 9, smallest first.
We add each pair.
We need 6.
3 and 3 give 6.
Yes!
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: 2(x + 3)(x − 3)/(x + 3)(x + 3).
(x + 3)/(x + 3) is 1.
Cross them out.
4 Write what is left
What is left is 2(x − 3)/(x + 3).
One more thing: the bottom of the question can't be 0.
So x can't be −3.
5 Check
Let's check with a number: x = 2.
First in the question.
We get −2/5.
Now in our answer.
We get −2/5 too.
Both are −2/5.
So 2(x − 3)/(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.