WEBVTT

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In this video, I'm going to show
you a technique called

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completing the square.

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And what's neat about this is
that this will work for any

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quadratic equation, and it's
actually the basis for the

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quadratic formula.

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And in the next video or the
video after that I'll prove

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the quadratic formula using
completing the square.

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But before we do that, we
need to understand even

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what it's all about.

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And it really just builds off
of what we did in the last

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video, where we solved
quadratics

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using perfect squares.

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So let's say I have the
quadratic equation x squared

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minus 4x is equal to 5.

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And I put this big space
here for a reason.

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In the last video, we saw
that these can be pretty

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straightforward to solve if
the left-hand side is a

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perfect square.

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You see, completing the square
is all about making the

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quadratic equation into a
perfect square, engineering

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it, adding and subtracting from
both sides so it becomes

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a perfect square.

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So how can we do that?

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Well, in order for this
left-hand side to be a perfect

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square, there has to be
some number here.

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There has to be some number here
that if I have my number

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squared I get that number, and
then if I have two times my

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number I get negative 4.

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Remember that, and I
think it'll become

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clear with a few examples.

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I want x squared minus 4x plus
something to be equal to x

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minus a squared.

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We don't know what a
is just yet, but we

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know a couple of things.

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When I square things-- so this
is going to be x squared minus

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2a plus a squared.

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So if you look at this pattern
right here, that has to be--

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sorry, x squared minus 2ax--
this right here has to be 2ax.

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And this right here would
have to be a squared.

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So this number, a is going to
be half of negative 4, a has

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to be negative 2, right?

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Because 2 times a is going
to be negative 4.

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a is negative 2, and if a is
negative 2, what is a squared?

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Well, then a squared is going
to be positive 4.

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And this might look all
complicated to you right now,

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but I'm showing you
the rationale.

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You literally just look at this
coefficient right here,

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and you say, OK, well what's
half of that coefficient?

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Well, half of that coefficient
is negative 2.

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So we could say a is equal to
negative 2-- same idea there--

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and then you square it.

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You square a, you
get positive 4.

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So we add positive 4 here.

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Add a 4.

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Now, from the very first
equation we ever did, you

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should know that you can never
do something to just one side

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of the equation.

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You can't add 4 to just one
side of the equation.

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If x squared minus 4x was equal
to 5, then when I add 4

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it's not going to be
equal to 5 anymore.

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It's going to be equal
to 5 plus 4.

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We added 4 on the left-hand side
because we wanted this to

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be a perfect square.

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But if you add something to the
left-hand side, you've got

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to add it to the right-hand
side.

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And now, we've gotten ourselves
to a problem that's

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just like the problems we
did in the last video.

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What is this left-hand side?

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Let me rewrite the
whole thing.

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We have x squared minus 4x
plus 4 is equal to 9 now.

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All we did is add 4 to both
sides of the equation.

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But we added 4 on purpose so
that this left-hand side

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becomes a perfect square.

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Now what is this?

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What number when I multiply it
by itself is equal to 4 and

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when I add it to itself I'm
equal to negative 2?

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Well, we already answered
that question.

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It's negative 2.

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So we get x minus 2 times
x minus 2 is equal to 9.

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Or we could have skipped this
step and written x minus 2

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squared is equal to 9.

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And then you take the square
root of both sides, you get x

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minus 2 is equal to
plus or minus 3.

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Add 2 to both sides, you get x
is equal to 2 plus or minus 3.

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That tells us that x could be
equal to 2 plus 3, which is 5.

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Or x could be equal to 2 minus
3, which is negative 1.

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And we are done.

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Now I want to be very clear.

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You could have done this without
completing the square.

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We could've started off
with x squared minus

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4x is equal to 5.

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We could have subtracted 5 from
both sides and gotten x

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squared minus 4x minus
5 is equal to 0.

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And you could say, hey, if I
have a negative 5 times a

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positive 1, then their product
is negative 5 and their sum is

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negative 4.

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So I could say this is x
minus 5 times x plus

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1 is equal to 0.

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And then we would say that x is
equal to 5 or x is equal to

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negative 1.

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And in this case, this actually
probably would have

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been a faster way to
do the problem.

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But the neat thing about the
completing the square is it

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will always work.

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It'll always work no matter what
the coefficients are or

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no matter how crazy
the problem is.

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And let me prove it to you.

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Let's do one that traditionally
would have been

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a pretty painful problem if
we just tried to do it by

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factoring, especially if we
did it using grouping or

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something like that.

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Let's say we had 10x squared
minus 30x minus

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8 is equal to 0.

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Now, right from the get-go, you
could say, hey look, we

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could maybe divide
both sides by 2.

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That does simplify
a little bit.

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Let's divide both sides by 2.

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So if you divide everything
by 2, what do you get?

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We get 5x squared minus 15x
minus 4 is equal to 0.

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But once again, now we have this
crazy 5 in front of this

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coefficent and we would have to
solve it by grouping which

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is a reasonably painful
process.

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But we can now go straight to
completing the square, and to

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do that I'm now going to divide
by 5 to get a 1 leading

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coefficient here.

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And you're going to see why this
is different than what

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we've traditionally done.

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So if I divide this whole thing
by 5, I could have just

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divided by 10 from the get-go
but I wanted to go to this the

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step first just to show
you that this really

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didn't give us much.

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Let's divide everything by 5.

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So if you divide everything by
5, you get x squared minus 3x

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minus 4/5 is equal to 0.

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So, you might say, hey, why did
we ever do that factoring

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by grouping?

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If we can just always divide by
this leading coefficient,

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we can get rid of that.

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We can always turn this into a 1
or a negative 1 if we divide

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by the right number.

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But notice, by doing that we
got this crazy 4/5 here.

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So this is super hard to do
just using factoring.

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You'd have to say, what two
numbers when I take the

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product is equal to
negative 4/5?

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It's a fraction and when I take
their sum, is equal to

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negative 3?

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This is a hard problem
with factoring.

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This is hard using factoring.

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So, the best thing to do is to
use completing the square.

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So let's think a little bit
about how we can turn this

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into a perfect square.

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What I like to do-- and you'll
see this done some ways and

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I'll show you both ways because
you'll see teachers do

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it both ways-- I like to get
the 4/5 on the other side.

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So let's add 4/5 to both
sides of this equation.

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You don't have to do it this
way, but I like to get the 4/5

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out of the way.

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And then what do we get
if we add 4/5 to both

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sides of this equation?

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The left-hand hand side of the
equation just becomes x

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squared minus 3x,
no 4/5 there.

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I'm going to leave a little
bit of space.

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And that's going to
be equal to 4/5.

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Now, just like the last problem,
we want to turn this

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left-hand side into the perfect
square of a binomial.

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How do we do that?

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Well, we say, well, what number
times 2 is equal to

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negative 3?

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So some number times
2 is negative 3.

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Or we essentially just take
negative 3 and divide it by 2,

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which is negative 3/2.

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And then we square
negative 3/2.

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So in the example, we'll
say a is negative 3/2.

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And if we square negative
3/2, what do we get?

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We get positive 9/4.

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I just took half of this
coefficient, squared it, got

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positive 9/4.

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The whole purpose of doing that
is to turn this left-hand

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side into a perfect square.

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Now, anything you do to one side
of the equation, you've

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got to do to the other side.

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So we added a 9/4 here, let's
add a 9/4 over there.

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And what does our
equation become?

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We get x squared minus 3x plus
9/4 is equal to-- let's see if

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we can get a common
denominator.

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So, 4/5 is the same
thing as 16/20.

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Just multiply the numerator
and denominator by 4.

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Plus over 20.

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9/4 is the same thing
if you multiply the

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numerator by 5 as 45/20.

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And so what is 16 plus 45?

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You see, this is kind of getting
kind of hairy, but

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that's the fun, I guess, of

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completing the square sometimes.

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16 plus 45.

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See that's 55, 61.

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So this is equal to 61/20.

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So let me just rewrite it.

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x squared minus 3x plus
9/4 is equal to 61/20.

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Crazy number.

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Now this, at least on
the left hand side,

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is a perfect square.

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This is the same thing as
x minus 3/2 squared.

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And it was by design.

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Negative 3/2 times negative
3/2 is positive 9/4.

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Negative 3/2 plus negative 3/2
is equal to negative 3.

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So this squared is
equal to 61/20.

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We can take the square root of
both sides and we get x minus

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3/2 is equal to the positive
or the negative

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square root of 61/20.

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And now, we can add 3/2 to both
sides of this equation

00:10:57.640 --> 00:11:03.600
and you get x is equal to
positive 3/2 plus or minus the

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square root of 61/20.

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And this is a crazy number and
it's hopefully obvious you

00:11:09.290 --> 00:11:11.430
would not have been able to-- at
least I would not have been

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able to-- get to this number
just by factoring.

00:11:15.250 --> 00:11:17.260
And if you want their actual
values, you can get your

00:11:17.260 --> 00:11:18.510
calculator out.

00:11:20.620 --> 00:11:22.510
And then let me clear
all of this.

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And 3/2-- let's do the plus
version first. So we want to

00:11:28.760 --> 00:11:33.710
do 3 divided by 2 plus the
second square root.

00:11:33.710 --> 00:11:35.050
We want to pick that little
yellow square root.

00:11:35.050 --> 00:11:46.480
So the square root of 61 divided
by 20, which is 3.24.

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This crazy 3.2464, I'll
just write 3.246.

00:11:52.760 --> 00:12:02.230
So this is approximately equal
to 3.246, and that was just

00:12:02.230 --> 00:12:03.110
the positive version.

00:12:03.110 --> 00:12:06.710
Let's do the subtraction
version.

00:12:06.710 --> 00:12:09.180
So we can actually put our
entry-- if you do second and

00:12:09.180 --> 00:12:11.535
then entry, that we want that
little yellow entry, that's

00:12:11.535 --> 00:12:12.465
why I pressed the
second button.

00:12:12.465 --> 00:12:16.130
So I press enter, it puts in
what we just put, we can just

00:12:16.130 --> 00:12:23.400
change the positive or the
addition to a subtraction and

00:12:23.400 --> 00:12:27.970
you get negative 0.246.

00:12:27.970 --> 00:12:33.800
So you get negative 0.246.

00:12:33.800 --> 00:12:38.200
And you can actually verify
that these satisfy our

00:12:38.200 --> 00:12:39.360
original equation.

00:12:39.360 --> 00:12:42.050
Our original equation
was up here.

00:12:42.050 --> 00:12:43.840
Let me just verify
for one of them.

00:12:47.400 --> 00:12:50.130
So the second answer on your
graphing calculator is the

00:12:50.130 --> 00:12:51.760
last answer you use.

00:12:51.760 --> 00:12:54.160
So if you use a variable answer,
that's this number

00:12:54.160 --> 00:12:55.160
right here.

00:12:55.160 --> 00:13:00.090
So if I have my answer squared--
I'm using answer

00:13:00.090 --> 00:13:02.380
represents negative 0.24.

00:13:02.380 --> 00:13:11.975
Answer squared minus 3 times
answer minus 4/5-- 4 divided

00:13:11.975 --> 00:13:16.030
by 5-- it equals--.

00:13:16.030 --> 00:13:18.490
And this just a little
bit of explanation.

00:13:18.490 --> 00:13:21.860
This doesn't store the entire
number, it goes up to some

00:13:21.860 --> 00:13:22.880
level of precision.

00:13:22.880 --> 00:13:24.910
It stores some number
of digits.

00:13:24.910 --> 00:13:28.930
So when it calculated it using
this stored number right here,

00:13:28.930 --> 00:13:32.240
it got 1 times 10 to
the negative 14.

00:13:32.240 --> 00:13:34.980
So that is 0.0000.

00:13:34.980 --> 00:13:37.100
So that's 13 zeroes
and then a 1.

00:13:37.100 --> 00:13:38.870
A decimal, then 13
zeroes and a 1.

00:13:38.870 --> 00:13:41.060
So this is pretty much 0.

00:13:41.060 --> 00:13:43.550
Or actually, if you got the
exact answer right here, if

00:13:43.550 --> 00:13:46.480
you went through an infinite
level of precision here, or

00:13:46.480 --> 00:13:49.050
maybe if you kept it in this
radical form, you would get

00:13:49.050 --> 00:13:52.390
that it is indeed equal to 0.

00:13:52.390 --> 00:13:55.300
So hopefully you found that
helpful, this whole notion of

00:13:55.300 --> 00:13:56.160
completing the square.

00:13:56.160 --> 00:13:58.670
Now we're going to extend it
to the actual quadratic

00:13:58.670 --> 00:14:01.510
formula that we can use, we
can essentially just plug

00:14:01.510 --> 00:14:03.610
things into to solve any
quadratic equation.