Factor completely 2x^2 - 4x - 30 Step by Step

Step by step

The question

Question: factor completely 2x² − 4x − 30.

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

2x² − 4x − 30

1 Common factor first

All the numbers can be divided by 2.

So the common factor is 2.

2, 4, 30: all in the 2 times table

We divide every term by it.

What is left goes in the bracket.

2x² ÷ 2 = x²

−4x ÷ 2 = −2x

−30 ÷ 2 = −15

So it is 2(x² − 2x − 15).

= 2(x² − 2x − 15)

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 −15, and they add to −2.

? × ? = −15

? + ? = −2

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

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

× is − → one +, one −; bigger one −

We list the pairs that make 15, smallest first.

We take the smaller from the bigger.

We need 2.

3 and 5 give 2.

Yes!

With the signs: 3 and −5.

1 × 15: 15 − 1 = 14

3 × 5: 5 − 3 = 2 ✓

So we get 2(x + 3)(x − 5).

= 2(x + 3)(x − 5)

3 Check each bracket

Brackets like x + 3 have just x and a number.

They can't be factored.

x + 3, x − 5: just x and a number

No bracket can be factored any more.

So we are done.

Careful!

Don't forget the common factor 2 in the answer.

Careful: not (x² − 2x − 15) ✗

4 Check

Let's check: we multiply it back out.

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

Check: (x + 3)(x − 5) = x² − 5x + 3x − 15

= x² − 2x − 15

Then the common factor times every part.

2(x² − 2x − 15) = 2x² − 4x − 30

We get 2x² − 4x − 30 again.

So it is right!

Let's look back.

First: common factor first.

Then: count the terms.

Then: check each bracket.

Last: check.

Answer