Simplify x/(x^2 - 4) + 1/(x + 2) Step by Step

Step by step

The question

Question: add x/(x² − 4) + 1/(x + 2).

xx² − 4 + 1x + 2

1 Factor the bottoms

x² − 4 is a square minus a square.

So it is (x + 2)(x − 2).

x² − 4 = (x + 2)(x − 2)

We write the bottoms as factors.

= x(x + 2)(x − 2) + 1(x + 2)

2 Find the common bottom

The bottoms are (x + 2)(x − 2) and (x + 2).

bottoms: (x + 2)(x − 2) and (x + 2)

The common bottom must have every factor of both bottoms, each one once.

It is (x + 2)(x − 2).

common bottom = (x + 2)(x − 2)

Careful!

We can't add the tops and add the bottoms.

Fractions need the same bottom first.

Careful: not x + 1x² − 2 + x ✗

3 Make the bottoms the same

The first fraction already has it.

The second fraction is missing (x − 2).

We multiply its top and bottom by (x − 2).

(x − 2)/(x − 2) is 1, so the value stays the same.

= x(x + 2)(x − 2) + (x − 2)(x + 2)(x − 2)

4 Add the tops

Now one fraction: (x + (x − 2)) over (x + 2)(x − 2).

= x + (x − 2)(x + 2)(x − 2)

5 Open the brackets

A bracket with a plus or nothing in front can just be removed.

( ) away

Now we have x + x − 2.

= x + x − 2(x + 2)(x − 2)

6 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

Each colour is one kind.

like terms = same letters,

same powers

x + x is 2x.

It is like 1 apple plus 1 apple: that is 2 apples.

x + x = 2x

So the expression is 2x − 2.

= 2x − 2(x + 2)(x − 2)

7 Cross out the same

Is a whole factor on top the same as one on the bottom?

No.

So nothing crosses out.

We leave the bottom as factors.

So the answer is (2x − 2)/(x + 2)(x − 2).

The bottoms can't be 0.

So x can't be −2 or 2.

x ≠ −2, 2

8 Check

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

First in the question.

1 over −3 plus 1/3 is 0.

Question: 11² − 4 + 11 + 2 = 1−3 + 13 = 0

Now in our answer.

We get 0 too.

Answer: 2(1) − 2(1 + 2)(1 − 2) = 0−3 = 0

Both are 0.

So (2x − 2)/(x + 2)(x − 2) is right!

Let's look back.

First: factor the bottoms.

Then: find the common bottom.

Then: make the bottoms the same.

Then: add the tops.

Then: open the brackets.

Then: group like terms.

Then: cross out the same.

Last: check.

Answer