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# Factoring quadratics by grouping

Sal factors 4y^2+4y-15 as (2y-3)(2y+5) by grouping. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to factor 4y squared
plus 4y, minus 15. And whenever you have an
expression like this, where you have a non-one coefficient
on the y squared, or on the second degree term-- it could
have been an x squared-- the best way to do this
is by grouping. And to factor by grouping we
need to look for two numbers whose product is equal to
4 times negative 15. So we're looking for two numbers
whose product-- let's call those a and b-- is going
to be equal to 4 times negative 15, or negative 60. And the sum of those two
numbers, a plus b, needs to be equal to this 4 right there. So let's think about all the
factors of negative 60, or 60. And we're looking for ones that
are essentially 4 apart, because the numbers are going
to be of different signs, because their product is
negative, so when you take two numbers of different signs and
you sum them, you kind of view it as the difference of
their absolute values. If that confuses you, don't
worry about it. But this tells you that the
numbers, since they're going to be of different size, their
absolute values are going to be roughly 4 apart. So we could try out things like
5 and 12, 5 and negative 12, because one has
to be negative. If you add these two you get
negative 7, if you did negative 5 and 12 you'd
get positive 7. They're just still
too far apart. What if we tried 6
and negative 10? Then you get a negative 4,
if you added these two. But we want a positive 4, so
let's do negative 6 and 10. Negative 6 plus 10
is positive 4. So those will be our two
numbers, negative 6 and positive 10. Now, what we want to do
is we want to break up this middle term here. The whole point of figuring out
the negative 6 and the 10 is to break up the 4y into
a negative 6y and a 10y. So let's do that. So this 4y can be rewritten as
negative 6y plus 10y, right? Because if you add
those you get 4y. And then the other sides of it,
you have your 4y squared, your 4y squared and then
you have your minus 15. All I did is expand this into
these two numbers as being the coefficients on the y. If you add these, you
get the 4y again. Now, this is where the
grouping comes in. You group the term. Let me do it in a
different color. So if I take these two guys,
what can I factor out of those two guys? Well, there's a common factor,
it looks like there's a common factor of 2y. So if we factor out 2y, we get
2y times 4y squared, divided by 2y is 2y. And then negative 6y divided
by 2y is negative 3. So this group gets factored
into 2y times 2y, minus 3. Now, let's look at this other
group right here. This was the whole point about
breaking it up like this. And in other videos I've
explained why this works. Now here, the greatest
common factor is a 5. So we can factor out a 5, so
this is equal to plus 5 times 10y, divided by 5 is 2y. Negative 15 divided
by five is 3. And so we have 2y times
2y minus 3, plus 5 times 2y minus 3. So now you have two terms,
and 2y minus 3 is a common factor to both. So let's factor out a 2y minus
3, so this is equal to 2y minus 3, times 2y, times
that 2y, plus that 5. There's no magic happening
here, all I did is undistribute the 2y minus 3. I factored it out of
both of these guys. I took it out of the
parentheses. If I distribute it in, you'd get
back to this expression. But we're done, we
factored it. We factored it into two
binomial expressions. 4y squared plus 4y, minus
15 is 2y minus 3, times 2y plus 5.