Factor 5x^2 - 13x - 6 Step by Step

Step by step

The question

Question: factor 5x² − 13x − 6.

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

5x² − 13x − 6

1 Find two numbers

The number in front of x² is 5.

So we multiply 5 by −6: that is −30.

We need two numbers that multiply to −30 and add to −13.

5 × (−6) = −30

? × ? = −30, ? + ? = −13

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 30, smallest first.

We take the smaller from the bigger.

We need 13.

2 and 15 give 13.

Yes!

With the signs: 2 and −15.

1 × 30: 30 − 1 = 29

2 × 15: 15 − 2 = 13 ✓

2 Split the middle

We write −13x as 2x − 15x.

It is the same amount.

= 5x² + 2x − 15x − 6

3 Group in pairs

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

= (5x² + 2x) + (−15x − 6)

From the first pair we take out x.

From the second pair we take out −3.

Both leave the same bracket.

(5x² + 2x) = x(5x + 2)

(−15x − 6) = −3(5x + 2)

Now both parts have 5x + 2 in them.

= x(5x + 2) − 3(5x + 2)

4 Same bracket out

5x + 2 is in both parts, so we take it out once.

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

ab + cb = b(a + c)

So we get (5x + 2)(x − 3).

= (5x + 2)(x − 3)

5 Check

Let's check: we multiply it back out.

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

Check: (5x + 2)(x − 3) = 5x² − 15x + 2x − 6

= 5x² − 13x − 6

We get 5x² − 13x − 6 again.

So (5x + 2)(x − 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