Simplify (x^2 - 9)/(x(x + 3)) × x/(x - 3)

Step by step

The question

Question: simplify (x² − 9)/x(x + 3) × x/(x − 3).

1 Factor every part

First the top of the first fraction: x² − 9.

x² − 9 is x² minus 3 squared.

A square minus a square is the plus bracket times the minus bracket.

So the top of the first fraction is (x + 3)(x − 3).

Now the bottom of the first fraction: x(x + 3).

It is already factored: x(x + 3) means x times (x + 3).

We just read the factors.

So the bottom of the first fraction is x(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.

Now every part is factored.

A bracket on any top and the same bracket on any bottom make 1.

(x + 3)/(x + 3) is 1.

Cross them out.

(x − 3)/(x − 3) is 1.

Cross them out.

x/x is 1.

Cross them out.

3 Multiply what is left

Top times top, bottom times bottom.

What is left is 1.

One more thing: the bottom of the question can't be 0.

So x can't be −3, 0 or 3.

4 Check

Let's check with a number: x = 2.

First in the question.

We get 1.

Now in our answer.

We get 1 too.

Both are 1.

So 1 is right!

Let's look back.

First: factor every part.

Then: cross out the same.

Then: multiply what is left.

Last: check.