Simplify 4/(x^2 - 4) - 1/(x

Step by step

The question

Question: work out 4/(x² − 4) − 1/(x − 2).

4x² − 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.

= 4(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 take away the tops and take away the bottoms.

Fractions need the same bottom first.

Careful: not 3x² − 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.

= 4(x + 2)(x − 2) − (x + 2)(x − 2)(x + 2)

4 Take away the tops

Now one fraction: (4 − (x + 2)) over (x + 2)(x − 2).

= 4 − (x + 2)(x + 2)(x − 2)

5 Open the brackets

A minus in front of a bracket changes the sign of every part inside.

−( + ) → −, −( − ) → +

Now we have 4 − x − 2.

= 4 − 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.

Plain numbers go together.

Each colour is one kind.

like terms = same letters,

same powers

We move like terms next to each other.

Each term takes its sign with it.

= −x + 4 − 2(x + 2)(x − 2)

4 − 2 is 2.

4 − 2 = 2

So the expression is −x + 2.

= −x + 2(x + 2)(x − 2)

7 Cross out the same

We write the top as factors too: −(x − 2).

= −~~(x − 2)~~(x + 2)~~(x − 2)~~

The same bracket on top and bottom is 1.

Cross them out.

What is left is −1/(x + 2).

= −1x + 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.

4 over −3 take away 1 over −1 is −1/3.

Question: 41² − 4 − 11 − 2 = 4−3 − 1−1 = −13

Now in our answer.

We get −1/3 too.

Answer: −1 + 2(1 + 2)(1 − 2) = 1−3 = −13

Both are −1/3.

So −1/(x + 2) is right!

Let's look back.

First: factor the bottoms.

Then: find the common bottom.

Then: make the bottoms the same.

Then: take away the tops.

Then: open the brackets.

Then: group like terms.

Then: cross out the same.

Last: check.

Answer