Factor 2x^2 - 5x - 3 Step by Step

Step by step

The question

Question: factor 2x² − 5x − 3.

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

2x² − 5x − 3

1 Find two numbers

The number in front of x² is 2.

So we multiply 2 by −3: that is −6.

We need two numbers that multiply to −6 and add to −5.

2 × (−3) = −6

? × ? = −6, ? + ? = −5

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 6, smallest first.

We take the smaller from the bigger.

We need 5.

1 and 6 give 5.

Yes!

With the signs: 1 and −6.

1 × 6: 6 − 1 = 5 ✓

2 Split the middle

We write −5x as x − 6x.

It is the same amount.

= 2x² + x − 6x − 3

3 Group in pairs

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

= (2x² + x) + (−6x − 3)

From the first pair we take out x.

From the second pair we take out −3.

Both leave the same bracket.

(2x² + x) = x(2x + 1)

(−6x − 3) = −3(2x + 1)

Now both parts have 2x + 1 in them.

= x(2x + 1) − 3(2x + 1)

4 Same bracket out

2x + 1 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 (2x + 1)(x − 3).

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

= 2x² − 5x − 3

We get 2x² − 5x − 3 again.

So (2x + 1)(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