Main content

### Course: Algebra (all content) > Unit 10

Lesson 16: Factoring polynomials with quadratic forms- Factoring quadratics: common factor + grouping
- Factoring quadratics: negative common factor + grouping
- Factor polynomials: quadratic methods
- Factoring two-variable quadratics
- Factoring two-variable quadratics: rearranging
- Factoring two-variable quadratics: grouping
- Factor polynomials: quadratic methods (challenge)
- Factoring quadratics with common factor (old)

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Factoring quadratics: negative common factor + grouping

Sal factors -12f^2-38f+22 as -2(2f-1)(3f+11). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- This isn't in the video, but I don't know where else to ask it:

While doing the exercise, I got -7x²-10x-3, and I simplified it like

-7x² - 10x - 3

-7x² (-7x - 3x) - 3

(-7x² -7x) (-3x-3)

-7x(x) -3(x)

x(-7x-3)

According to the test, this was wrong. Can someone explain why?(11 votes)- When you factored, you dropped you constant term. Here is the correct way to do it:

-7x² - 10x - 3

− (7x² + 10x + 3)

− (7x² + 7x + 3x + 3)

− [(7x(x + 1) + 3(x + 1)]

− (x+1)(7x + 3)(22 votes)

- i got -2(-3x-11)(2x-1) is that correct?(5 votes)
- Close, you made two small errors.

The variable was an f and you changed it to a x. No big deal, but you the letter in the answer shoudl still be f.

and second, your (-3x-11) shoudl be (+3f-11) the sign on the 3 should be positive.

Check you work and see where you missed the sign.

Good luck(7 votes)

- This might be a weird question but how would you solve for what f is??(5 votes)
- The problem in the video is an expression, not an equation. We can't solve expressions.

However, if the problem had started out as:

-12f^2-38f+22 = 0, then we could solve for "f".

This is a quadratic equation.

1) Factor (as shown in the video): -2(2f-1)(3f+11) = 0

2) Then we use the zero product rule that let's us split the factors into individual equations:

2f-1=0 and 3f+11=0.

Note, we ignore the -2 factor because it will not create a solution.

3) We then solve each individual equation:

2f-1=0 creates f=1/2

3f+11=0 creates f= -11/3

For more info on solving - see https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:factored-form-quadratics/v/zero-product-property(4 votes)

- is there any other way to solve the problem besides writing it out?(2 votes)
- i have a question that is (x+2)^2 - (x+2) - 42 does this go to 4x^2 - 2x - 42? and if so what do i do then?(3 votes)
- You can substitute a plain x for (x+2) and factor first to make it easier, but then you have to replace it back. So it becomes x^2-x-42. Then facor that: (x-7)(x+6)then plug the (x+2) back in for the x.

(x+2-7)(x+2+6)= (x-5)(x+8)ANSWER

If you multiply (x+2)^2 first it equals x^2+4x+4 not 4x^2 so be careful and remember to foil. This method is more work, but it would be x^2+4x+4-x-2-42

after simplifying it would be x^2+3x-40. Then factor to (x+8)(x-5) ANSWER(4 votes)

- At1:22, Mr.Khan says to try a few numbers that add to 19 and multiply to -66. This works for small numbers, but what about if you have a really LARGE quadratic to factor? Then it would take a very long time to find the 2 numbers. I have been searching up methods, and the quickest one I found was the prime factorization method. However, this is still taking too long. My quadratic I am trying to factor is 84x^2+181x-90. Multiplying the 84 and -90 gets me to 7260, but it is taking me too long to figure out what 2 numbers multiply to that and add to 181. I am looking for a method I can use to factor these really fast because in my math counts competition I cant spend forever on one question.

Thanks!(3 votes) - Why does a times b have to be -66 and a + b have to be 19?(3 votes)
- This is just a little tool we use to make sure that the our numbers will fit neatly inside each other in the next step. Sal goes in depth on this in some earlier videos on factoring: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:intro-factoring/v/factors-and-divisibility-in-algebra?modal=1(2 votes)

- Is there any way to do this a little faster? It takes a lot of time to do, and I have a time limit on all of my tests and quizzes.(2 votes)
- Just practice a lot! Once you really understand it, (you still have to do each step) you can do the steps faster. From experience, I can say that after doing these for about a week or so, I've really gotten the hang of it, from doing about 20+ factoring polynomials a day! Hope this helps!(4 votes)

- I got -2(-3f-11)(-2f+1). Does anyone care to share why this differs from Sal's presumably optimal answer?(2 votes)
- You have extra factors of -1 applied to both your binomials. This is why your signs are different. If you factor out those -1's, you can get to Sal's version

-3f-11 = -1(3f+11)

-2f+1 = -1(2f-1)

You now have: -2(-1)(-1)(3f+11)(2f-1)

Multiply the 3 constants to get Sal's final version of: -2(3f+11)(2f-1)

To get your version, you would have had to make a conscious decision to factor out -3f when Sal factored out +3f, and then you did the same late by factor out a -2f when Sal did +2f. That's what caused the extra factors of -1 to creep in.

To avoid this situation in the future, only factor out a negative value if you need to change the highest degree term to be positive. They were already positive in this problem after the GCF of -2 had been factored out.

Hope this helps.(3 votes)

- Why did he factor out a negative two at0:21? I have a test coming up and I have no idea what Im doing.(2 votes)
- He factors out a negative two because -12f^2, -38f, and 22 all can be factored out by two, and he factors by negative two because then the leading coefficient will not be negative, making the question easier.(2 votes)

## Video transcript

We need to factor negative 12f
squared minus 38f, plus 22. So a good place to start is just
to see if, is there any common factor for all three
of these terms? When we look at them,
they're all even. And we don't like a negative
number out here. So let's divide everything, or
let's factor out a negative 2. So this expression right here is
the same thing as negative 2 times-- what's negative 12f
squared divided by negative 2? It's positive 6f squared. Negative 38 divided by negative
2 is positive 19, so it'll be positive 19f. And then 22 divided by negative
22-- oh, sorry, 22 divided by negative
2 is negative 11. So we've simplified it a bit. We have the 6f squared
plus 19f, minus 11. We'll just focus on that
part right now. And the best way to factor this
thing, since we don't have a 1 here as the coefficient
on the f squared, is to factor it by grouping. So we need to look for two
numbers whose product is 6 times negative 11. So two numbers, so a times b,
needs to be equal to 6 times negative 11, or negative 66. And a plus b needs to
be equal to 19. So let's try a few
numbers here. So let's see, 22, I'm just
thinking of numbers that are roughly 19 apart, because
they're going to be of different signs. So 22 and 3, I think
will work. Right. If we take 22 times negative 3,
that is negative 66, and 22 plus negative 3 is
equal to 19. And the way I kind of got pretty
close to this number is, well, you know, they're
going to be of different signs, so the positive versions
of them have to be about 19 apart, and
that worked out. 22 and negative 3. So now we can rewrite this 19f
right here as the sum of negative 3f and 22f. That's the same thing as 19f. I just kind of broke it apart. And, of course, we have the
6f squared and we have the minus 11 here. Now, you're probably saying, hey
Sal, why did you put the 22 here and the negative
3 there? Why didn't you do it the
other way around? Why didn't you put the 22 and
then the negative 3 there? And my main motivation for doing
it, I like to put the negative 3 on the same side with
the 6 because they have the common factor of the 3. I like to put the 22 with the
negative 11, they have the same common factor of 11. So that's why I decided
to do it that way. So now let's do the grouping. And, of course, you can't forget
this negative 2 that we have sitting out here
the whole time. So let me put that negative 2
out there, but that'll just kind of hang out for awhile. But let's do some grouping. So let's group these
first two. And then we're going to group
this-- let me get a nice color here-- and then we're going
to group this second two. Well, that's almost an
identical color. Let me do it in this
purple color. And then we can group that
second two right there. So these first two, we could
factor out a negative 3f, so it's negative 3f times-- 6f
squared divided by negative 3f is negative 2f. And then negative 3f divided
by negative 3f is just positive f. Actually, a better way to start,
instead of factoring out a negative 3f, let's just
factor out 3f, so we don't have a negative out here. We could do it either way. But if we just factor
out a 3f, 6f squared divided by 3f is 2f. And then negative 3f divided
by 3f is negative 1. So that's what that
factors into. And then that second part, in
that dark purple color, can factor out an 11. And if we factor that out, 22f
divided by 11 is 2f, and negative 11 divided by
11 is negative 1. And, of course, once again,
you have that negative 2 hanging out there. Now, inside the parentheses,
we have two terms, both of which have 2f minus
1 as a factor. So we can factor that out. This whole thing is just an
exercise in doing the reverse distributive property,
if you will. So let's factor that out, so
you have 2f minus 1, times this 3f, and then times
that plus 11. Let me do that in the
same shade of purple right over there. And you know, you can distribute
it if you like. 2f minus 1 times 3f will give
you this term, 2f f minus 1 times 11 will give
you that term. And we can't forget that we
still have that negative 2 hanging out outside. I want to change the
colors on it. And we're done factoring it. Negative 12f squared minus 38f,
plus 22 is negative 2 times 2f minus 1, times
3f plus 11.