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This video is going to be about factor the difference of two perfect squares.
So,
I want to start out by multiplying
two binomials together
and show you how we get to
the problem we're trying to solve.
So let's multiply x plus 7 times x minus 7.
So we FOIL these. We get x times x will give me x squared.
We're going to get x times negative 7
is negative
7x.
Positive 7 times x is positive
7x
and 7 times negative 7
is negative 49. Now watch what happens when I simplify this by
combining the middle terms.
I'm going to be x squared,
and the middle terms are the sign, I've got a negative 7x
and positive 7x.
So they're going to cancel each other. They're going to equal zero,
and then we're just going to end up with the negative
49.
x squared
is a perfect square, and 49 is perfect square, and
when we subtract one number from another, we say we have the difference of those
numbers.
So this is the difference
of two squares,
and this is a pattern you're going to see in a number of problems. Teachers for some
reason love to get thing as a problem on tests, so what I want to do
is give you some problems where we start out with something like this and then
realize how we get back to
the factored version.
So let's start with the problem
x squared
minus
9, and we want to factor that
into 2 binomials.
So we know that the first term in each binomial is going to be an x,
because we've got an x squared here.
looking at this negative 9, we realize 9 is a perfect square
and there's no middle term.
Well when we have a difference of two perfect squares
and no middle term
basically to solve it
all we have to do is take the square root of the second term we have,
put a 3 here
and a 3 here,
make one of these positive
and one of them negative.
And we can multiply it back just to make sure we're right.
So x times x is x squared,
x times negative 3 is negative 3x,
3 times x is positive 3x,
3 times negative 3 is negative 9, and you can see already that these two are
going to cancel
and we'll get x squared
minus 9.
Let's try a couple more.
So let's look at this one.
Now about
25
a squared
minus
16.
Examining it,
we realize that 25 is a perfect square
and a squared is a perfect square, so 25a squared is a perfect square.
16 is a perfect square, there's no middle term and we're taking the difference of
those two squares.
So really what we have to do
put in our parentheses, take the square root of the first term,
which is 5a,
make that the first term of each of our binomials,
put in one positive sign
and one negative sign
and then take the square root of the second term,
which is 4, the square root of 16 is 4,
and this is going to be our answer.
5a plus 4, 5a minus 4.
Notice the pattern of the answers: you've got basically two identical answers,
two identical binomials, except
you've got a positive sign in one and a negative sign in the other.
Again,we can check this my multiplying it back.
5a times 5a
is 25a squared.
5a times negative 4 is negative 20a.
4 times 5a is positive 20a.
And 4 times negative 4 is negative 16.
These terms in the middle cancel.
You're going to get
25a squared
minus 16.
Okay, one more.
So let's take
8
a squared
minus
18.
Now,
8 is not a perfect square and neither is 18.
But, before we just decide that we can't do this,
let's try simplifying this by factoring out a common factor. Both these numbers
are even,
so why don't we try
factoring out a 2.
Dividing the 8 by 2,
I'm going to get a 4, so that'll be 4a squared,
minus... dividing 18 by 2 I'll get a 9.
So now looking at what I have inside parentheses, this is 4a
squared,
which is a perfect square,
and 9, which is a perfect square.
So this binomial
can be factored.
So let's keep our 2,
and
get our two
final binomials.
So the square root of 4 is 2, and the square root of a squared is a.
So I'll have a 2a,
for both of these
a plus sign and a minus sign, the square root of 9 is 3.
So there's my answer...
2 times 2a plus 3 times 2a minus 3.
Checking it,
I'll multiply these two binomials together first
so 2a times 2a is
4a squared, 2a times negative 3
is negative 6a,
3 times 2a is positive 6a,
and 3 times negative 3 is negative 9.
I can cross that these two because they cancel each other, negative 6a and
positive 6a.
And now distributing my 2,
I get
8a squared...
2 tmes negatively 9 is negative 18.
And that's exactly the same problem I began with.
So this factorization
is going to work.
So reviewing the steps...
first, to see if you can use this method,
see if you have two terms.
If they're both perfect squares
then
you're ready to go on. If they're not both perfect squares,
see if you can
divided out, if you can factor out some number
which will leave you with two perfect squares.
Next, make sure that you're dealing with a difference... this sign has to be
negative.
If it's positive, stop right there,
you can't go on.
So you've got two perfect squares, you've got the difference of them,
take the square root of the first one,
that's going to be the first term in each of your binomials. Take the
square root of the second term, that'll be the second term in each of your binomials.
Put in a plus sign for one of the binomials,
negative sign for the other binomial,
and you're done.
And that's really all there is to it.
Practice a bunch of these. I guarantee you'll see them on tests.
Take care,
see you next time.