Simplify (x^2 - 4)/(3x) ÷ (x + 2)/(6x) Step by Step

Step by step

The question

Question: simplify (x² − 4)/3x ÷ (x + 2)/6x.

x² − 43x ÷ x + 26x

1 Flip and multiply

Dividing by a fraction is the same as times its flip.

Keep the first fraction.

Change divide to times.

Turn the second fraction upside down.

÷ x + 26x → × 6xx + 2

So we work out (x² − 4)/3x × 6x/(x + 2).

= x² − 43x × 6xx + 2

2 Factor every part

First the top of the first fraction: 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 of the first fraction is (x + 2)(x − 2).

= (x + 2)(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 every part is factored.

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

= ~~(x + 2)~~(x − 2)3x × 6x~~(x + 2)~~

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

Cross them out.

x/x is 1.

Cross them out.

6 and 3 can both be divided by 3.

6 becomes 2, 3 becomes 1.

4 Multiply what is left

Top times top, bottom times bottom.

What is left is 2(x − 2).

= 2(x − 2)

One more thing: the bottoms 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, flipped.

We get −2.

Question: 1² − 43(1) × 6(1)1 + 2 = −33 × 63 = −2

Now in our answer.

We get −2 too.

Answer: −2

Both are −2.

So 2(x − 2) is right!

Let's look back.

First: flip and multiply.

Then: factor every part.

Then: cross out the same.

Then: multiply what is left.

Last: check.

Answer