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# Factoring quadratics: negative common factor + grouping

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.