Factor completely 3x^2 + 3x - 18 Step by Step

Step by step

The question

Question: factor completely 3x² + 3x − 18.

Completely means: until no part can be factored any more.

3x² + 3x − 18

1 Common factor first

All the numbers can be divided by 3.

So the common factor is 3.

3, 3, 18: all in the 3 times table

We divide every term by it.

What is left goes in the bracket.

3x² ÷ 3 = x²

3x ÷ 3 = x

−18 ÷ 3 = −6

So it is 3(x² + x − 6).

= 3(x² + x − 6)

2 Count the terms

The bracket has 3 terms.

With 3 terms we look for two numbers that multiply to c and add to b.

We need two numbers.

They multiply to −6, and they add to 1.

? × ? = −6

? + ? = 1

They multiply to a minus number, so one number is plus and one is minus.

They add to a plus number, so the bigger one is plus.

× is − → one +, one −; bigger one +

We list the pairs that make 6, smallest first.

We take the smaller from the bigger.

We need 1.

2 and 3 give 1.

Yes!

With the signs: −2 and 3.

1 × 6: 6 − 1 = 5

2 × 3: 3 − 2 = 1 ✓

So we get 3(x − 2)(x + 3).

= 3(x − 2)(x + 3)

3 Check each bracket

Brackets like x − 2 have just x and a number.

They can't be factored.

x − 2, x + 3: just x and a number

No bracket can be factored any more.

So we are done.

Careful!

Don't forget the common factor 3 in the answer.

Careful: not (x² + x − 6) ✗

4 Check

Let's check: we multiply it back out.

Each part of one bracket times each part of the other.

Check: (x − 2)(x + 3) = x² + 3x − 2x − 6

= x² + x − 6

Then the common factor times every part.

3(x² + x − 6) = 3x² + 3x − 18

We get 3x² + 3x − 18 again.

So it is right!

Let's look back.

First: common factor first.

Then: count the terms.

Then: check each bracket.

Last: check.

Answer