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Use completing the square to
write the quadratic equation y

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is equal to negative 3x squared,
plus 24x, minus 27 in

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vertex form, and then
identify the vertex.

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So we'll see what vertex form
is, but we essentially

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complete the square, and we
generate the function, or we

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algebraically manipulate it so
it's in the form y is equal to

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A times x minus B
squared, plus C.

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We want to get the equation
into this form right here.

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This is vertex form
right there.

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And once you have it in vertex
form, you'll see that you can

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identify the x value of the
vertex as what value will make

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this expression equal to 0.

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So in this case it would be B.

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And the y value of the vertex,
if this is equal to 0, then

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the y value is just
going to be C.

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And we're going to see that.

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We're going to understand why
that is the vertex, why this

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vertex form is useful.

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So let's try to manipulate
this equation to get

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it into that form.

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So if we just rewrite it, the
first thing that immediately

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jumps out at me, at least, is
that all of these numbers are

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divisible by negative 3.

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And I just always find it
easier to manipulate an

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equation if I have a 1
coefficient out in front of

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the x squared.

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So let's just factor
out a negative 3

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right from the get-go.

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So we can rewrite this as y is
equal to negative 3 times x

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squared, minus 8x-- 24 divided
by negative 3 is

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negative 8-- plus 9.

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Negative 27 divided by negative
3 is positive 9.

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Let me actually write the
positive 9 out here.

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You're going to see in a second
why I'm doing that.

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Now, we want to be able to
express part of this

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expression as a perfect
square.

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That's what vertex
form does for us.

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We want to be able to express
part of this expression as a

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perfect square.

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Now how can we do that?

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Well, we have an x
squared minus 8x.

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So if we had a positive 16
here-- because, well, just

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think about it this way, if we
had negative 8, you divide it

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by 2, you get negative 4.

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You square that, it's
positive 16.

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So if you had a positive
16 here, this would

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be a perfect square.

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This would be x minus
4 squared.

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But you can't just willy-nilly
add a 16 there, you would

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either have to add a similar
amount to the other side, and

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you would have to scale it by
the negative 3 and all of

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that, or, you can just subtract
a 16 right here.

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I haven't changed
the expression.

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I'm adding a 16, subtracting
a 16.

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I've added a 0.

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I haven't it changed it.

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But what it allows me to do is
express this part of the

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equation as a perfect square.

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That right there is
x minus 4 squared.

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And if you're confused, how
did I know it was 16?

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Just think, I took negative
8, I divided by 2, I

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got negative 4.

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And I squared negative 4.

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This is negative 4 squared
right there.

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And then I have to subtract that
same amount so I don't

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change the equation.

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So that part is x
minus 4 squared.

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And then we still have this
negative 3 hanging out there.

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And then we have negative 16
plus 9, which is negative 7.

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So we're almost there.

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We have y equal to negative 3
times this whole thing, not

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quite there.

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To get it there, we just
multiply negative 3.

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We distribute the negative 3 on
to both of these terms. So

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we get y is equal to negative
3 times x, minus 4 squared.

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And negative 3 times negative
7 is positive 21.

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So we have it in our vertex
form, we're done with that.

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And if you want to think about
what the vertex is, I told you

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how to do it.

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You say, well, what's the
x value that makes

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this equal to 0?

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Well, in order for this term to
be 0, x minus 4 has to be

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equal to 0.

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x minus 4 has to be equal to 0,
or add 4 to both sides. x

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has to be equal to 4.

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And if x is equal to 4, this is
0, this whole thing becomes

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0, then y is equal to 21.

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So the vertex of this parabola--
I'll just do a

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quick graph right here-- the
vertex of this parabola occurs

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at the point 4, 21.

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So I'll draw it like this.

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Occurs at the point.

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If this is the point 4, if this
right here is the-- so

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this is the y-axis, that's
the x-axis-- so this is

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the point 4, 21.

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Now, that's either going to be
the minimum or the maximum

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point in our parabola, and to
think about whether it's the

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minimum or maximum point, think
about what happens.

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Let's explore this equation
a little bit.

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This thing, this x minus 4
squared is always greater than

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or equal to 0.

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Right?

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At worst it could be 0, but
you're taking a square, so

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it's going to be a non-negative
number.

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But when you take a non-negative
number, and then

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you multiply it by negative 3,
that guarantees that this

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whole thing is going to be
less than or equal to 0.

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So the best, the highest, value
that this function can

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attain, is when this
expression right

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here is equal to 0.

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And this expression is equal
to 0 when x is equal

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to 4 and y is 21.

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So this is the highest value
that the function can attain.

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It can only go down
from there.

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Because if you shift the x
around 4, then this expression

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right here will become, well,
it'll become non-zero.

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When you square it, it'll
become positive.

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When you multiply it
by negative 3,

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it'll become negative.

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So you're going to take a
negative number plus 21, it'll

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be less than 21, so your
parabola is going

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to look like this.

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Your parabola is going
to look like that.

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And that's why vertex
form is useful.

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You break it up into the part
of the equation that changes

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in value, and say,
well, what's its

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maximum value attained?

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That's the vertex.

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That happens when
x is equal to 4.

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And you know its y value.

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And because you have a negative
coefficient out here

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that's a negative 3, you know
that it's going to be a

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downward opening graph.

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If that was a positive 3, then
this thing would be, at

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minimum, 0 and it would be
an upward opening graph.

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