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 Question from melissa, a student: how do you factor this completely: 2x^2-3x-5

We have two responses for you

Hi Melissa,

I am going to try a similar problem: Factor completely 3x2 - x - 2.

I would first look for any common factors. There are none so next I would try to factor the trinomial as

3x2 - x - 2 = (?x + ?)(?x + ?)

where all the ?s are integers and I don't know their values or even their signs. Now look at expanding the right side. First you multiply the two x terms to get the x2 term on the left which is 3x2. Since 3 can only be factored as 3 × 1 I must have

3x2 - x - 2 = (3x + ?)(x + ?)

Again look at expanding the right side but this time notice that the product of the two ?s must be -2 so one is positive and the other is negative. Hence there are only four possibilities.

3x2 - x - 2 = (3x + 1)(x - 2)
3x2 - x - 2 = (3x - 1)(x + 2)
3x2 - x - 2 = (3x + 2)(x - 1) or
3x2 - x - 2 = (3x - 2)(x + 1)

If you expand each of the right sides you know that the x2 term and the constant term match the left side but only one gives the x term as -x. Which one is it?

Penny

Hi Melissa.

Watch how I factor this example and you can follow the same steps to factor your question.

Factor 3x2 + 4x - 7.

First I make my parentheses and put the x's in place:

(  x       )  (  x       )

Next I look at the number in front of the squared term. It is a 3, so I need to find two numbers that multiply to make 3. That's either 1 and 3 or -1 and -3. I'll try 1 and 3.

(1x      ) (3x       )

Now I look at the last digit. It is a 7, so two numbers that multiply to make 7 are 1 and 7. I'll try those.

(1x     7) (3x     1)

The middle term of the quadratic is +4x. So I want to organize this to make +4x by moving things around and changing signs as needed.

outside terms: 1x(1) = 1x.
inside terms: 7(3x) = 21x.

There's no way I can make +4x by combining 1x and 21x. So I'll switch the 7 and the 1:

(1x     1) (3x     7)

outside terms: 1x(7) = 7x.
inside terms: 1(3x) = 3x.

Now we are close, because 7x - 3x makes the 4x I'm looking for. So I just need to set the minus and plus properly in the parentheses now:

(1x - 1) (3x + 7)

To check, I use FOIL to multiply it out:

1x(3x) -1(3x) + (1x)(7) -1(7)
=3x2 - 3x + 7x - 7
=3x2 + 4x - 7. That's what I wanted.

So 3x2 + 4x - 7 factors into (1x - 1) (3x + 7).

There are several more questions in our archive that deal with factoring. You can look at those if you want to see more.

Now you try the same steps to find the factors of 2x2 - 3x - 5, Melissa.

Cheers,
Stephen La Rocque.

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