Step by step
The question
Question: simplify (x² − 9)/(x² + 3x).
A fraction with x in it is called a rational expression.
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.
So the top is (x + 3)(x − 3).
2 Factor the bottom
We look at the bottom: x² + 3x.
Every part has x.
We take x out.
What is left goes in the bracket.
So the bottom is x(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.
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(x + 3).
(x + 3)/(x + 3) is 1.
Cross them out.
4 Write what is left
What is left is (x − 3) over x.
One more thing: the bottom of the question can't be 0: we can't divide by 0.
The bottom x is 0 when x is 0.
x + 3 is 0 when x is −3.
The answer keeps this rule, even after crossing out.
So x can't be −3 or 0.
5 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 −1/2.
Now in our answer.
We get −1/2 too.
Both are −1/2.
So (x − 3)/x 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.