WEBVTT

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In the completing the square
video I kept saying that all

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the quadratic equation is
completing the square

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as kind of a short cut
of completing square.

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And I was under the impression
that I had done this

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proof already but now I
realize that I haven't.

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So let me prove the quadratic
equation to you, by

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completing the square.

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So let's say I have a
quadratic equation.

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I guess a quadratic equation is
actually what you're trying to

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solve, and what a lot of people
call the quadratic equation is

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actually the quadratic formula.

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But anyway I don't want to get
caught up in terminology.

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But let's say that I have
a quadratic equation that

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says ax squared plus bx
plus c is equal to 0.

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And let's just complete
the square here.

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So how do we do that?

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Well let's subtract c from both
sides so we get ax squared plus

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the bx is equal to minus c.

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And just like I said in the
completing the square video

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I don't like having this
a coefficient here.

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I like just having one
coefficient on my x squared

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term so let me divide
everything by a.

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So I get x squared plus b/a
x is equal to-- you have

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to divide both sides
by a --minus c/a.

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Now we are ready to
complete the square.

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What was completing the square?

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Well it's somehow adding
something to this expression so

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it has the form of something
that is the square

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of an expression.

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What do i mean by that?

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Well, I'll do a
little aside here.

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if I told you that x plus
a squared, that equals

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x squared plus two ax
plus a squared, right?

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So if I can add something here
so that this left hand side

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this expression looks like
this, then I could

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go the other way.

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I can say this is going to be
x plus something squared.

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So what do I have to
add on both sides?

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If you watched the completing
the square video this should be

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hopefully intuitive for you.

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What you do is you say well
this b/a, this corresponds to

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the 2a term, so a is going to
be half of this, is going to

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be half of this coefficient.

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That would be the a.

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And then what I need
to add is a squared.

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So I need to take half of
this and then square it and

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then add it to both sides.

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Let me do that in a
different color.

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Do it in this magenta.

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So I'm going to take half of
this-- I'm just completing

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square, that's all I'm doing,
no magic here --so

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plus half of this.

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Well half of that
is b/2a right?

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You just multiply by 1/2.

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And I have to square it.

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Well if I did it to the left
hand side of the equation, I

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have to do it to the
right hand side.

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So plus b/2a squared.

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And now I have this left hand
side of the equation in the

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form that it is the square of
an expression that is

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x plus something.

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And what is it?

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Well that's equal to-- let me
switch colors again --what's

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the left hand side of
this equation equal to?

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And you can just use this
pattern and go to the left.

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It's x plus what?

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Well we said a, you can do one
of two ways. a is 1/2 of this

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coefficient or a is the square
root of this coefficient or

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since we didn't even square it
we know that this

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is a. b/2a is a.

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So this is the same thing as x
plus b over 2a everything

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squared, and then that equals--
let's see if we can simplify

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this or make this a little
bit cleaner --that equals--

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See, if I were to have a common
denominator-- I'm just doing a

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little bit of algebra here
--see, when I square this it's

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going to be 4a squared--
let me let me write this.

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This is equal to b
squared over 4a squared.

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Right?

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And so if I have to add these
two fractions, let me make

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this equal to 4a squared.

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Right?

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And if the denominator is
4a squared, what does

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the minus c/a become?

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I See if I multiply the
denominator by 4a, I have to

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multiply the numerator by 4a.

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So this becomes
minus 4ac, right?

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And then b squared over
4a squared, well that's

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just still b squared.

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I'm just doing a little
bit of algrebra.

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Hopefully I'm not
confusing you.

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I just expanded this.

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I just took the square of this,
b squared over 4a squared.

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And then I added this to this,
I got a common denominator.

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And minus c/a is the same thing
as minus 4ac over 4a squared.

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And now we can take the
square root of both

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sides of this equation.

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And this should hopefully
start to look a little

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bit familiar to you now.

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So let's see, so we get x.

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So if we take the square root
of both sides of this equation

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we get x plus b/2a is equal to
the square root of this-- let's

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take the square root of the
numerator and the demoninator.

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So the numerator is-- I'm going
to put the b squared first, I'm

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just going to switch this
order, it doesn't matter --the

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square root of b squared
minus 4ac, right?

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That's just the numerator.

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I just the square root of it,
and we have to get the square

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root of the denominator too.

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What's the square
roof of 4a squared?

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Well it's just 2a, right?

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2a.

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And now what do we do?

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Oh, it's very important!

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When we're taking the square
root, it's not just the

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positive square root.

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It's the positive or
minus square root.

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We saw that couple of times
when we did the-- and you could

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say it's a plus or minus here
too, but if you look plus or

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minus on the top and a plus or
minus on the bottom, you can

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just write it once on the top.

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I'll let you think about why
you only have to write it once.

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If you had a negative an a
plus, or negative and a plus

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sometimes cancel out, or a
negative and a negative,

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that's the same thing as
just having a plus on top.

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Anyway, I think you get that.

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And now we just have to
subtract b/2a from both sides.

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and we get, we get-- and this
is the exciting part --we get x

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is equal to minus be over to 2a
plus or minus this thing, so

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minus b squared minus 4ac,
all of that over 2a.

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And we already have a common
denominator, so we can

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just add the fractions.

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So we got --and I'm going to do
this in a vibrant bold-- I

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don't know maybe not so much
bold, well green color --so we

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get x is equal to, numerator,
negative b plus or minus square

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root of b squared minus
4ac, all of that over 2a.

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And that is the famous
quadratic formula.

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So, there we go we proved it.

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And we proved it just from
completing the square.

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I hope you found that
vaguely interesting.

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See in the next Video.