>> Okay. So, let's see
why the Quadratic Formula?
So, our Quadratic Formula
said, "If a x squared plus b x,
plus c is zero with a
[inaudible]", all right?
"Then something", right, let's
see if we can figure out the why
for the "then something."
So, I would like to solve
something like this,
and I have a method to do this,
it's called complete a square,
and this looks like it's
going to be horrible, right?
So, if I was to try to complete
the square on this thing,
what's the first
step I would do?
>> Subtract c.
>> Okay. Subtract the
c on the other side,
so I get a x squared plus b x --
>> Need a space.
>> Okay, I need a blank, right?
>> Yeah.
>> Is?
>> Negative c.
>> Negative c plus blank?
>> Yeah. And then
divide b x by 2.
>> Wait.
>> Or b?
>> You're jumping
ahead of yourself.
What's the other problem?
You guys see it?
Why aren't I ready
to divide b by 2?
Why wasn't I really ready
to draw blanks in here yet?
>> Because it's -- a is not a 1?
>> A to divide.
>> Yeah, I got to divide through
by the leading coefficient,
right?
I have to make a 1 to be
able to complete the square.
So, I got to divide
through by a here.
Does that seem horrible?
>> [inaudible] seems horrible.
>> Yeah, it kind of is
horrible, so what do I get?
I get x squared --
>> Plus b x over a.
>> Plus -- okay, I'm going
to write this a funny way.
This should be b
x over a, right?
I'm going to write it as b
over a times x. You
guys okay with that?
>> Why?
>> X divided by a.
>> Because multiplication
doesn't care
about the order, right?
This is an -- a b times x, times
a one over a in whatever order.
You guys see that?
All right, b times
x, times one over a.
>> It doesn't matter
what order -- ?
Yeah, okay.
>> And what I really decided to
do is write this as b times one
over a, and group
those two together,
and then multiply by x.
>> Yeah, okay, that makes sense.
>> You guys all see
those are the same thing?
>> Yeah.
>> The reason for me writing
it this way is I want to keep
that kind of, flavor of
the coefficient on x is b
over a. Okay, then
there was this blank
which thankfully didn't
have anything in it,
otherwise I would have
to divide it by a, right?
And c, what's -- ?
>> Negative c over a.
>> Okay, so I get
negative c over a,
and then there's a blank.
So, I really wasn't
ready for those yet.
Maybe I'll say, I didn't put
them in and now I put them
in after I put -- divided by
a. Okay, so now I want to think
about this thing as x plus
something squared, right?
>> Yeah.
>> Okay, so what's the
something that goes here?
>> B over a times
x, divided by 2.
>> Do I need the x?
>> No. Because remember, I
take the coefficient on x,
right, that's b over a?
I divide it by 2.
What's b over a divided by 2?
>> B over a --
>> B over -- ?
>> A time 1/2?
>> A times two --
>> That's exactly
what I [inaudible] --
>> 2 a.
>> So, this is b over 2 a.
Trust me on that for a moment,
you might to fiddle
with fractions to get
that one down, but it's there.
Okay, so I got b
over 2 a-squared,
what goes in the blanks now?
>> Or no, b --
>> B over 2 a squared?
>> B over 2 a all
squared, right?
What goes in the other blank?
>> B over 2 a squared as well?
>> That same thing, right?
Okay, so what's the right-hand
side of that equation?
>> B over 2 a squared minus
c. Oh, no, minus c over a.
>> So, I've got b over 2 a, all
squared minus c over a, right?
You guys cool with that?
>> Okay, now what?
At this stage I would
normally like -- ?
>> Try to factor
out roots or any --
>> I would try to -- ?
>> Extract the roots.
>> Extract the roots, right?
Okay, but right now that
right-hand side totally sucks.
So, can I make the
right-hand side any better?
So, basically, I'm just looking
to fiddle with this thing
to maybe make it even one
fraction, so that it's not
such a pain in the butt.
>> You could try to set
the bases to each other,
or you have to multiply
b over 2 a out.
>> Okay. So, I've
got to multiply b
over 2 a times itself.
So, I get -- ?
>> 2 b --
>> B -- ?
>> B squared.
>> Over?
>> 2 a squared.
No, 4 a --
>> 2 a all squared,
which is four a squared.
>> A squared, yeah.
>> And then I'm subtracting --
>> It's c, oh you'll
have to make --
>> Okay. So, I've
got c over a, right?
So, let me just write c over
a. Now, what do I want to do
with these two to make them --
so I can stick them together?
>> Yeah, I should try to
find a common bond, right?
What's the common bond
between 4 a squared and a?
>> 4 a squared.
>> 4 a squared, right?
This thing's already
got an a in it.
>> So you would --
>> In fact, it's
got two of them.
>> So, you have to multiply c --
>> By?
>> 4 a.
>> 4 a.
>> And the top as well.
>> And the top as well, right?
I need to multiply by 1.
Okay, so I've got b squared --
>> Over --
>> Minus --
>> Four a squared minus --
>> Wait.
>> Wait, b squared minus c.
>> Four --
>> Oh, a c.
>> A, c.
>> A, c is the standard
form, right?
>> And then they're all
over a, 4 a squared.
>> You guys seeing that part?
>> Yeah.
>> That looks familiar, right?
That's the discriminant
in the Quadratic
Formula actually, right?
>> Oh --
>> Cool, right?
It's not magic, it
comes from somewhere.
So, the left-hand side, which
I lost a bit ago was x plus b
over a squared, right?
>> 2 a squared.
>> Oh, yeah, so now
what do I do?
>> You can extract the root.
>> Okay, so what's my
first self-extracting root?
>> Do minus square
root on both sides.
>> Plus or minus --
>> Okay, so we plus, minus
square root both sides.
So, I get x plus b over a
-- oops, b over 2 a, right?
This whole thing is --
>> The square root of b --
>> The square root --
>> Or plus or minus.
>> Yeah, plus or minus.
>> Plus or minus the square root
of b squared minus 4
a c, over 4 a squared.
>> So --
>> Now what?
>> They need to --
>> [inaudible] but
the 4 has got to --
>> Subtract b, and
subtract b over 2 a.
>> Yeah, I want to -- that thing
out of there, so I subtract b
over 2 a. So, I end up
here with x is negative --
>> B --
>> B over 2 a, plus or
minus the square root
of b squared minus 4
a c, over 4 a squared.
>> Cool.
>> You guys see that?
>> Play you're just adding in --
>> Play, and then you rewrite
this as, okay, x is negative b
over 2 a, plus or minus --
and then, if I'm squaring a
fraction I can square at the top
and square at the bottom.
So, you get the square root
of b squared minus 4 a c,
over the square root
of 4 a squared.
>> Which is 2 a.
>> Oh, I already knew the
square root of that --
>> Yeah.
>> Because I squared something
and got it earlier, right?
So, I'm going to
replace this with 2 a.
>> And then they combine.
>> And then they combine, so
I get x is negative b plus
or minus the square root of b
squared minus 4 a c, over 2 a.
>> All right, that was
pretty cool [inaudible].
>> Okay? So, what do I
expect you to take from this?
I expect you be able to do -- ?
>> Oh, that [inaudible] from --
>> No, but I do expect
you to know
that the Quadratic Formula comes
from complete the square, right?
It's not magic, it
really just comes
out of completing the square.
And you should know how
to complete the square,
because if you don't you just
memorize [inaudible] cool?
>> Yeah.
>> So, who's the first dude
that would like solve -- ?
