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CCSS Math: HSA.SSE.A.2, HSA.SSE.B.3, HSF.IF.C.8

Factor 25x squared
minus 30x plus 9. And we have a leading
coefficient that's not a 1, and it doesn't look like
there are any common factors. Both 25 and 30 are divisible by
5, but 9 isn't divisible by 5. We could factor
this by grouping. But if we look a little
bit more carefully here, see something interesting. 25 is a perfect square, and 25x
squared is a perfect square. It's the square of 5x. And then nine is also
a perfect square. It's the square of
3, or actually, it could be the square
of negative 3. This could also be the
square of negative 5x. Maybe, just maybe this
could be a perfect square. Let's just think
about what happens when we take the perfect square
of a binomial, especially when the coefficient on
the x term is not a 1. If we have ax plus b
squared, what will this look like when we expand
this into a trinomial? Well, this is the same
thing as ax plus b times ax plus b, which is the same
thing as ax times ax. Ax times ax is a squared x
squared plus ax times b, which is abx plus b times
ax, which is another. You You could call
it bax or abx, plus b times b, so plus b squared. This is equal to a squared
x squared plus-- these two are the same term--
2abx plus c squared. This is what happens when
you square a binomial. Now, this pattern seems
to work out pretty good. Let me rewrite our
problem right below it. We have 25x squared
minus 30x plus 9. If this is a
perfect square, then that means that the a squared
part right over here is 25. And then that means
that the b squared part-- let me do this in
a different color-- is 9. That tells us that a
could be plus or minus 5 and that b could
be plus or minus 3. Now let's see if this gels
with this middle term. For this middle
term to work out-- I'm trying to look for good
colors-- 2ab, this part right over here, needs to
be equal to negative 30. Or another way-- let me write
it over here-- 2ab needs to be equal to negative 30. Or if we divide
both sides by 2, ab needs to be equal
to negative 15. That tells us that the
product is negative. One has to be positive,
and one has to be negative. Now, lucky for us the
product of 5 and 3 is 15. If we make one of them positive
and one of them negative, we'll get up to negative 15. It looks like things
are going to work out. We could select a is
equal to positive 5, and b is equal to negative 3. Those would work out to ab
being equal to negative 15. Or we could make a is
equal to negative 5, and b is equal to positive 3. Either of these will work. If we factor this
out, this could be either a is negative--
let's do this first one. It could either be a
is 5, b is negative 3. This could either be
5x minus 3 squared. a is 5, b is negative 3. It could be that. Or you could have-- we could
switch the signs on the two terms. Or a could be negative 5,
and b could be positive 3. Or it could be negative
5x plus 3 squared. Either of these
are possible ways to factor this term out here. And you say wait, how
does this work out? How can both of these
multiply to the same thing? Well, this term, remember,
this negative 5x plus 3, we could factor
out a negative 1. So this right here is the same
thing as negative 1 times 5x minus 3, the whole
thing squared. And that's the same thing
as negative 1 squared times 5x minus 3 squared. And negative 1 squared
is clearly equal to 1. That's why this and
this are the same thing. This comes out to the same
thing as 5x minus 3 squared, which is the same thing
as that over there. Either of these are
possible answers.