Factor 6x^2 + 7x - 3 Step by Step

Step by step

The question

Question: factor 6x² + 7x − 3.

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

6x² + 7x − 3

1 Find two numbers

The number in front of x² is 6.

So we multiply 6 by −3: that is −18.

We need two numbers that multiply to −18 and add to 7.

6 × (−3) = −18

? × ? = −18, ? + ? = 7

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 18, smallest first.

We take the smaller from the bigger.

We need 7.

2 and 9 give 7.

Yes!

With the signs: −2 and 9.

1 × 18: 18 − 1 = 17

2 × 9: 9 − 2 = 7 ✓

2 Split the middle

We write 7x as −2x + 9x.

It is the same amount.

= 6x² − 2x + 9x − 3

3 Group in pairs

We put the first two together, and the last two together.

= (6x² − 2x) + (9x − 3)

From the first pair we take out 2x.

From the second pair we take out 3.

Both leave the same bracket.

(6x² − 2x) = 2x(3x − 1)

(9x − 3) = 3(3x − 1)

Now both parts have 3x − 1 in them.

= 2x(3x − 1) + 3(3x − 1)

4 Same bracket out

3x − 1 is in both parts, so we take it out once.

What is left, 2x + 3, goes in the other bracket.

ab + cb = b(a + c)

So we get (3x − 1)(2x + 3).

= (3x − 1)(2x + 3)

5 Check

Let's check: we multiply it back out.

Each part of the first bracket times each part of the second.

Check: (3x − 1)(2x + 3) = 6x² + 9x − 2x − 3

= 6x² + 7x − 3

We get 6x² + 7x − 3 again.

So (3x − 1)(2x + 3) is right!

Let's look back.

First: find two numbers.

Then: split the middle.

Then: group in pairs.

Then: same bracket out.

Last: check.

Answer