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CCSS Math: HSA.REI.B.4, HSA.REI.B.4b, HSA.SSE.B.3, HSA.SSE.B.3a, HSF.IF.C.8, HSF.IF.C.8a

We're asked to solve for s. And we have s squared minus
2s minus 35 is equal to 0. Now if this is the first time
that you've seen this type of what's essentially a quadratic
equation, you might be tempted to try to solve for s using
traditional algebraic means, but the best way to solve this,
especially when it's explicitly equal to 0, is to
factor the left-hand side, and then think about the fact that
those binomials that you factor into, that they have
to be equal to 0. So let's just do that. So how can we factor this? We've seen it in several ways. I'll show you the standard
we've been doing it, by grouping, and then there's a
little bit of a shortcut when you have a 1 as a coefficient
over here. So when you do something by
grouping, when you factor by grouping, you think about two
numbers whose sum is going to be equal to negative 2. So you think about two numbers
whose sum, a plus b, is equal to negative 2 and whose product
is going to be equal to negative 35. a times b is equal
to negative 35. So if the product is a negative
number, one has to be positive, one has
to be negative. And so if you think about it,
ones that are about two apart, you have 5 and negative 7,
I think that'll work. 5 plus negative 7 is equal
to negative 2. So to factor by grouping, you
split this middle term. We can split this into a--
let me write it this way. We have s squared, and then this
middle term right here, I'll do it in pink. This middle term right there I
can write it as plus 5s minus 7s and then we have
the minus 35. And then, of course, all
of that is equal to 0. Now, we call it factoring by
grouping because we group it. So we can group these first two
terms. And these first two terms, they have a common
factor of s. So let's factor that out. You have s times s plus 5. That's the same thing as
s squared plus 5s. Now, in these second two terms
right here, you have a common factor of negative 7, so
let's factor that out. So you have negative
7 times s plus 5. And, of course, all of
that is equal to 0. Now, we have two terms here,
where both of them have s plus 5 as a factor. So we can factor that out. So let's do that. So you have s plus 5 times
this s right here, right? S plus 5 times s will
give you this term. And then you have minus
that 7 right there. I undistributed the s plus 5. And then this is going
to be equal to 0. Now that we've factored it, we
just have to think a little bit about what happens
when you take the product of two numbers? I mean, s plus 5 is a number. s minus 7 is another number. And we're saying that the
product of those two numbers is equal to zero. If ever told you that I had
two numbers, if I told you that I had the numbers a times
b and that they equal to 0, what do we know about either
a or b or both of them? Well, at least one of them has
to be equal to 0, or both of them have to be equal to 0. So, the fact that this number
times that number is equal to zero tells us that either s plus
5 is equal to 0 or-- and maybe both of them-- s minus
7 is equal to 0. I'll do that in just green. And so you have these two
equations, and actually, we could say and/or. It could be or/and, either way,
and both of them could be equal to 0. So let's see how we can
solve for this. Well, we can just subtract
5 from both sides of this equation right there. And so you get, on the left-hand
side, you have s is equal to negative 5. That is one solution to the
equation, or you can add 7 to both sides of that equation, and
you get s is equal to 7. So if s is equal to negative 5,
or s is equal to 7, then we have satisfied this equation. We can even verify it. If you make s equal to negative
5, you have positive 25 plus 10, which is minus 35. That does equal zero. If you have 7, 49 minus 14
minus 35 does equal zero. So we've solved for s. Now, I mentioned there's
an easier way to do it. And when you have something like
this, where you have 1 as the leading coefficient,
you don't have to do this two-step factoring. Let me just show
you an example. If I just have x plus a
times x plus b, what is that equal to? x times x is x squared,
x times b is bx. a times x is plus ax. a times b is ab. So you get x squared plus--
these two can be added-- plus a plus bx plus ab. And that's the pattern that
we have right here. We have 1 as a leading
coefficient here, we have 1 as a leading coefficient here. So once we have our two
numbers that add up to negative 2, that's our a plus
b, and we have our product that gets to negative 35, then
we can straight just factor it into the product of
those two things. So it will be-- or the product
of the binomials, where those will be the a's and the b's. So we figured it out. It's 5 and negative 7. 5 plus negative 7
is negative 2. 5 times negative 7
is negative 35. So we could have just straight
factored at this point. 2, well, actually this
was the case of s. So we could have factored it
straight to the case of s plus 5 times s minus 7. We could have done that straight
away and would've gotten to that right there. And, of course, that whole
thing was equal to zero. So that would've been a little
bit of a shortcut, but factoring by grouping is a
completely appropriate way to do it as well.