Simplify (x^2 - 4)/(x(x + 2)) Step by Step

Step by step

The question

Question: simplify (x² − 4)/x(x + 2).

A fraction with x in it is called a rational expression.

x² − 4x(x + 2)

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.

69 = 2 × 33 × 3 = 23

1 Factor the top

We look at the top: x² − 4.

= x² − 4

x² − 4 is x² minus 2 squared.

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

a² − b² = (a + b)(a − b)

So the top is (x + 2)(x − 2).

= (x + 2)(x − 2)

2 Factor the bottom

We look at the bottom: x(x + 2).

= x(x + 2)

It is already factored: x(x + 2) means x times (x + 2).

We just read the factors.

x(x + 2) = x × (x + 2)

So the bottom is x(x + 2).

= x(x + 2)

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.

Careful: only whole brackets cross out

Now we write the fraction with the factors: (x + 2)(x − 2)/x(x + 2).

= ~~(x + 2)~~(x − 2)x~~(x + 2)~~

(x + 2)/(x + 2) is 1.

Cross them out.

4 Write what is left

What is left is (x − 2) over x.

= x − 2x

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

So x can't be −2 or 0.

x ≠ −2, 0

5 Check

Let's check with a number: x = 1.

First in the question.

We get −1.

Question: 1² − 41(1 + 2) = −33 = −1

Now in our answer.

We get −1 too.

Answer: 1 − 21 = −1

Both are −1.

So (x − 2)/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.

Answer