Which expression is equivalent to (x^2 - 25)/(x - 5)?

Step by step

The question

Question: which expression is equivalent to (x² − 25)/(x − 5)?

Equivalent means: the same value for every x.

A fraction with x in it is called a rational expression.

There are four choices.

First we simplify it, the normal way.

Then we pick.

1 Factor the top

We look at the top: x² − 25.

Factor means: write it as a times, with brackets.

x² − 25: 25 is 5 times 5, so it is x² minus 5 squared.

A square minus a square is the plus bracket times the minus bracket.

So the top is (x + 5)(x − 5).

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: (x + 5)(x − 5)/(x − 5).

(x − 5)/(x − 5) is 1.

Cross them out.

3 Write what is left

What is left is x + 5.

One more thing: the bottom of the question can't be 0: we can't divide by 0.

x − 5 is 0 when x is 5.

The answer keeps this rule, even after crossing out.

So x can't be 5.

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 x + 5 is right!

5 Pick the choice

Our answer is x + 5.

That is choice C.

6 Why not the others

A is x − 25.

Wrong: crossed out the x's: only whole brackets cancel.

B is x − 5.

Wrong: x² − 25 = (x + 5)(x − 5): the (x − 5) is the one that cancels.

D is x² − 5.

Wrong: divided −25 by 5 only.

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.