Simplify 6/(x^2 + 5x + 6) + 2/(x + 2) Step by Step

Step by step

The question

Question: add 6/(x² + 5x + 6) + 2/(x + 2).

6x² + 5x + 6 + 2x + 2

1 Factor the bottoms

For x² + 5x + 6 we need two numbers that multiply to 6 and add to 5.

So it is (x + 2)(x + 3).

x² + 5x + 6 = (x + 2)(x + 3)

We write the bottoms as factors.

= 6(x + 2)(x + 3) + 2(x + 2)

2 Find the common bottom

The bottoms are (x + 2)(x + 3) and (x + 2).

bottoms: (x + 2)(x + 3) and (x + 2)

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

It is (x + 2)(x + 3).

common bottom = (x + 2)(x + 3)

Careful!

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

Fractions need the same bottom first.

Careful: not 8x² + 6x + 8 ✗

3 Make the bottoms the same

The first fraction already has it.

The second fraction is missing (x + 3).

We multiply its top and bottom by (x + 3).

(x + 3)/(x + 3) is 1, so the value stays the same.

= 6(x + 2)(x + 3) + 2(x + 3)(x + 2)(x + 3)

4 Add the tops

Now one fraction: (6 + 2(x + 3)) over (x + 2)(x + 3).

= 6 + 2(x + 3)(x + 2)(x + 3)

5 Open the brackets

The 2 in front of the bracket multiplies every part inside.

a(b + c) = ab + ac

So we write 2 times each part.

= 6 + 2(x) + 2(3)(x + 2)(x + 3)

2 times x is 2x.

2 times 3 is 6.

= 6 + 2x + 6(x + 2)(x + 3)

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.

= 2x + 6 + 6(x + 2)(x + 3)

6 + 6 is 12.

6 + 6 = 12

So the expression is 2x + 12.

= 2x + 12(x + 2)(x + 3)

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 + 12)/(x + 2)(x + 3).

The bottoms can't be 0.

So x can't be −3 or −2.

x ≠ −3, −2

8 Check

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

First in the question.

6/12 plus 2/3 is 7/6.

Question: 61² + 5(1) + 6 + 21 + 2 = 612 + 23

= 76

Now in our answer.

We get 7/6 too.

Answer: 2(1) + 12(1 + 2)(1 + 3) = 1412 = 76

Both are 7/6.

So (2x + 12)/(x + 2)(x + 3) 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