Step by step
The question
Question: simplify (6x² + 9x)/3x.
A fraction with x in it is called a rational expression.
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.
Next we do the same with x and brackets.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Factor the top
We look at the top: 6x² + 9x.
Factor means: write it as a times, with brackets.
6 and 9 can both be divided by 3.
Every part has x.
We take 3x out.
What is left goes in the bracket.
So the top is 3x(2x + 3).
2 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.
A number check: 2 plus 3, all over 3, is 5/3.
Crossing out the 3s would give 2.
Wrong!
Now we write the fraction with the factors: 3x(2x + 3)/3x.
x/x is 1.
Cross them out.
So 3 goes into both.
We divide both by 3.
3 becomes 1, 3 becomes 1.
3 Write what is left
What is left is 2x + 3.
One more thing: the bottom of the question can't be 0: we can't divide by 0.
These come from the question, so they stay.
So x can't be 0.
4 Check
Let's check with a number: x = 2.
We pick a small number that makes no bottom 0.
First in the question.
We get 7.
Now in our answer.
We get 7 too.
Both are 7.
So 2x + 3 is right!
5 Pick the choice
Our answer is 2x + 3.
That is choice A.
6 Why not the others
B is 6x² + 3.
Wrong: divided only the 9x by 3x.
C is 2x² + 3x.
Wrong: divided by 3, not by 3x.
D is 2x + 9x.
Wrong: divided only the 6x² by 3x.
Let's look back.
First: factor the top.
Then: cross out the same.
Then: write what is left.
Then: check.
Then: pick the choice.
Last: why not the others.