>> What we want to
talk about is 7.3.
And really this is going to be
the technique to end techniques
for solving quadratics.
So this is using the
Quadratic Formula.
Ok, so the problem pattern
that goes with this,
is this little theorem.
Which really says, if you add
ax squared plus bx plus c is 0.
The thing I'm stating
here, by the way,
is really the quadratic formula.
So if we have one of these
guys, right, this is kind
of the quadratic equals
0, then there's an answer
for this that's relatively easy.
x is given by?
Do any of you guys
know this formula?
Have you heard this
thing before?
Maybe not.
So given by negative b, plus
or minus the square root
of b squared minus
4ac all over 2a.
>> I remember that.
>> Some of you probably remember
this from someplace else.
There's a little song to the
tune of Pop Goes the Weasel
that you may learn to memorize.
>> Is it about a dance club,
or dance, or something?
>> What?
>> One was about a dance.
>> Oh yeah, there's
a bunch of these.
They're all horrible.
I have too much dignity,
and that's saying a lot
that I have too much dignity.
>> Put it on the internet, Joe.
>> I have too much
dignity to sing the song.
>> Sing the song.
>> No. I'm not singing the song.
So if some of you would like to
look up the song and learn it,
I would recommend that.
>> Just say it.
Who cares?
I want to hear one.
>> I said it as I
was writing it down.
It also goes to Pop
Goes the Weasel.
I'm not singing it.
>> You don't have to sing it.
>> Yeah, that.
Negative b, plus or
minus the square root
of b squared minus
4ac all over 2a.
Put that in your head
to Pop Goes the Weasel.
>> You're trying so
hard not to sing it.
>> I know.
>> Sing it.
>> No, no.
Dignity. Right up here.
>> How does [inaudible].
>> Ew, no.
I'd be horrible.
No, Pop Goes the Weasel.
So keep in mind, this only
works when you have if this.
Right? The quadratic formula
doesn't work when some stuffs
on the left and some
stuffs on the right.
It's all got to be on the left,
that said it doesn't
really care about signage.
So it doesn't care whether you
move stuff to the left hand side
or the right hand side.
>> It'll be the same.
>> It'll be the same either way.
>> And it has to equal 0.
>> Yeah, it's got to
equal 0, and it's got
to all be on one side.
Cool? All right.
Do you want to do an example
or do you want to learn the y?
>> Example.
>> Example?
>> The song.
[Laughter]
>> The song was not
on the table,
it's still not on the table.
I'm still not going to sing it.
Ok, so if we have x squared
minus 5x minus 6 is 0.
[ Inaudible ]
>> So you do x equals.
>> So the first thing you
should do upon seeing one
of these is what?
[ Inaudible ]
So you could complete the
square on this one, right?
Why is this one a good -
wait, why is this one a good
or bad candidate for
completing the square?
>> 5x is odd so [inaudible].
>> The 5's odd, so
identical to fraction.
Right? But I don't have to
divide through by an a yet.
Right? So at the very least,
this is only a medium
bad complete the square.
But I want to use my new toy.
The first thing you should
do when you're trying
to solve a quadratic
formula kind
of problem is move
everything to one side.
Ok, it's already done that.
Check. Everything's on one side.
Now the next thing I should do
is sort out what are a, b and c.
So what are the a, b and c here?
>> a is 1.
>> a is 1.
Good, b is?
>> Negative 5?
>> Very nice.
You've got to keep the
negatives with these.
So this one is negative 5.
And the c is?
>> Negative 6.
>> Negative 6.
Then what?
>> Then you file it in?
>> Yeah, then you just
write this formula down.
So x is, and I'm supposed
to have negative b,
so that's negative, what's b?
>> Negative 5.
>> Very nice, negative 5.
Then I'm supposed to have
plus or minus the square root.
What's b again?
>> Negative 5.
>> So I get negative
5, wow, horrible 5.
5 squared minus 4.
>> 4 times the 1 times.
>> Times 1, times negative 6.
>> Ok.
>> Divided by 2 by.
>> Divided by 2 times 1.
>> These are often
times a little bit ugly.
Right? This is the price you're
going to pay for not having
to remember how to complete
the square, or for not having
to fight with fractions.
You're going to have
to pay a little bit
of simplification price here.
Ok, do you see that?
You're also going to pay a
penalty if you can't factor.
Right? Ok, do you see that?
If I can obviously
factor this thing--
[ Inaudible ]
Then I should probably do that.
Right? Ok, you see that?
Factoring here would be way
easier, maybe, than doing this.
So try to factor for
a second, look at it,
think about whether complete
the square is the right idea.
If it isn't, then
go onto this thing.
So the Quadratic Formula should
be your kind of last resort.
But it works all of the time.
So if the other techniques
aren't going to give it
to you, this one will.
>> Every time?
>> Yeah, this one
will work every time.
>> The price you might
pay for getting it
to work is some kind
of weirdness in here.
This might be hard to deal
with and there's a thing
that might happen that
we'll come across later,
where we could maybe
take the square root
of a negative number.
And that's not, not
working, we just have
to learn to deal with it.
We'll get there, I promise.
So with this one, the next thing
I should do is try to simplify.
Right? Ok, so what things
can I simplify easily?
>> The 2 and the 5?
>> I know, let's see.
2 times 1 is 2 and
negative, negative 5 is?
>> 5.
>> 5. And a plus or minus
will just chill, right?
Ok, now do I know any
of the bits in there?
>> You know 25.
>> 25?
>> Ok, so this is all
I'm going to square root.
Negative 5 squared is 25.
Is it positive or negative?
>> Positive.
>> Positive.
Very nice.
Good.
>> Minus 4?
[ Inaudible ]
>> Ok, see that?
I like to multiply
these things together.
So what's negative 4
times 1 times negative 6?
>> Positive 24.
>> 24.
>> 24, yeah.
>> Positive 24.
Ok, so now.
>> That's a square root right?
>> Yeah, that's a square root.
So now, order of operations says
which of these pieces
has to happen first?
>> Add in the 24 and 25?
>> Ok, I have to add
in the 24 and the 25.
So I get 5 plus or minus the
square root of something over 2.
What's the something?
>> 49.
>> 49, right.
And then I can replace
this whole business.
The square root of 49
is something I know.
>> 7.
>> It's 7.
So I get 5 plus or minus, wait
I know the square root of 49.
>> 7.
>> Right, that's 7.
And then this is over?
>> 2?
>> 2.
>> It starts to be -- Would
you get two different answers?
>> There's two different
answers.
S x is five plus
7 over 2 or x is?
>> 5 minus 7 over 2?
>> 5 minus 7 over 2.
So this guy gets you what?
>> 6 and negative 1?
>> 6 or x is negative 1.
Ok. What does this mean?
>> That your parabola.
>> That my parabola.
Right.
>> The parabola.
>> The parabola.
>> The parabola that
I started with,
my x squared minus 5x minus 6.
>> Is at positive
6 and negative 1.
>> And it's an x-axis.
>> It's an x-axis at -
>> Negative 1
>> Negative 1 and -
>> Positive.
>> Positive 6.
Which means that I could have
factored this silly thing.
Ok, you see that?
>> So do you do AC method on it?
>> I think Completing the
Square is an easier method
than this one.
>> It depends.
So Completing the Square is an
easy method when this term is 1
and this coefficient is even.
If those two things
aren't happening,
then probably Quadratic
Formula is easier.
This one we should
have factored though.
Not using the AC method,
but just saying ok,
I had to multiply the negative
6 and add the negative 5.
That's negative 6
and positive 1.
Ok, you see that?
[ Inaudible ]
So that would be x minus 6 times
x plus 1 is 0 when factored.
>> That sounds simple.
>> And I would get x is
6 or x is negative 1.
You guys see that?
So either way, there's
nothing really new here,
but this method works
when I can't factor.
It also works when I
can't Complete the Square.
Well, it works when
Completing the Square would be
inconveniently difficult.
I can always Complete
the Square,
it just sometimes
gets really horrible.
Ok, you see that?
When we do the y part of this,
we'll see that this thing
actually is Completing
the Square.
It's the same formula.
>> Ok, so like when I'm
Completing the Square it's going
to give you a bunch
of messy fractions,
I'll use this [inaudible].
>> When Completing
the Square is going
to give you some horrible
number, like when I have
to divide by, here,
let's make one up.
We can just make up
an example and try it.
Ok, so what makes Completing
the Square horrible?
I'm just looking
to make an example.
I've got x squared and
x something equals 0.
[ Inaudible ]
Oops, that's supposed to be a 5.
Let's make it a 3
because I can draw a 3.
[ Inaudible ]
[ Laughter ]
Ok. Do you guys agree that this
would be particularly horrible
to Complete the Square on?
>> Yes.
>> Because, first, I would have
to divide the whole
thing through by 7.
And then I'd have to
divide three 7th's by two
and then square that and that
gets confusing and nasty.
But this is an easy
Quadratic Formula problem
because all I have to
do is get everything
to one side, check, right?
Then what?
>> Then you can just plug in.
You have to find a, b and c.
>> Ok, so what are a, b and c?
>> S7, 3 and 1?
>> So a is 7, b is 3, c is 1.
>> And you can just plug them
into the Quadratic Formula.
>> Ok, so let's try it.
So x is?
>> Negative 7?
>> Negative.
>> No, 3. It's b.
>> It says b. So it's negative
3 plus or minus the square root?
>> Of b squared.
>> Which is?
>> 9. I guess you can --
>> 3 squared.
So I would advise, personally,
that you write 3 squared.
>> So you don't square
root it twice.
>> Yeah, so you don't
accidentally square it twice,
and so that you can check, once
you've gone through the formula,
whether you've got
everything in the right spot.
So 4, and then?
>> Times 7, times 1.
>> Times 7, times 1.
>> Divided by -
>> And then divided by?
>> 2 times 7.
>> Twice 7.
Oh, cool, good.
This is going to highlight
another thing that can happen.
So, let's see, what do I --
the price I paid for not
Completing the Square was
that I now have to
simplify some stuff, right?
Ok, so what stuff do I aim at?
What stuff's easy to simplify?
>> The 3 squared?
>> Ok, so, let's see,
I've got negative 3 plus
or minus the square root of
>> 9.
>> 9 minus -
>> 28.
>> 28 divided by -
>> 14
>> 14. Ok, so x is
negative 3 plus
or minus the square
root of what over 14?
What's 9 minus 28?
>> Negative 19?
>> Negative 19.
What do we do with that?
>> Divide.
>> Do you guys see the problem?
>> The negative or
the square root --
>> Yeah, I'm asking
what number do I square
to get a negative number.
Right?
>> There isn't one.
>> There isn't one.
[ Inaudible ]
There just isn't one.
So the answer to this is -
[ Inaudible ]
No real solutions.
Good. What does this mean?
There is a parabola, right?
And I was wondering if that
parabola hit the x axis.
Can you guys draw a
picture of a parabola
that doesn't hit the x axis?
>> Just up one [inaudible]?
>> Something like that.
So what have I learned about
the parabola 7x squared plus 3x
plus 1?
>> That it does not
hit the x axis?
>> Yeah, it just
doesn't hit the x axis.
That's not bad, it
just happened,
I just learned something.
You guys see that?
Ok, so when you get a negative
under the square root, in fact,
that thing, this part
of the Quadratic Formula,
this guy right here.
What did I just find?
>> If it's negative, it's
not hitting the x axis.
>> Yeah, if it's negative,
there's no real solutions.
Right? This thing is so
special that we give it a name,
we call it the discriminant.
[ Inaudible ]
And that's spelled
wrong, probably.
>> If we can clearly see that
it's going to be negative,
do we have to keep going?
>> No.
>> Cool.
>> So one thing you could
learn from the beginning is
to check this thing to quick
fill out b squared minus 4ac
because if that's
negative, you can just write
down no real solutions.
If it's 0, what do you write?
>> Under.
>> Undefined.
No?
>> What's the square root of 0?
>> 0.
>> 0.
>> 0, right, So do
you guys see what --
It hit's the x axis once
because I get a negative b plus
or minus 0 over 2a.
>> Ok.
>> You guys see that?
So it's possible that I
get one real solution.
>> So it's just a line?
>> Or, no, it's not just a line.
>> It just hits the x axis.
>> Can I just -
>> No it just hits
the x axis, yeah.
>> Yeah, if I want
to draw a parabola
that just hits the x axis once,
it looks like this, right?
>> Right, yeah.
>> So this picture goes
with discriminant equals 0.
Yep. That's one real solution.
If the discriminant is -- this
picture goes with discriminant?
>> No real solutions?
>> No solutions?
>> No real solutions.
Ok. But the discriminant
there was?
>> Negative 19?
>> Negative specifically.
The discriminant
was smaller than 0.
And the other picture.
What's the other picture guys?
>> Undefined?
>> Where it hits it twice?
>> Yeah, that's where
it hits twice, right?
That's where the
discriminant is?
[Inaudible] It's
bigger than 0, yeah.
It's positive.
And that gives me two
real solutions, right?
You guys see that?
This is my one real
solution there.
These are my two
real solutions there.
And this one, I don't need
to circle anything right,
because there aren't
any real solutions.
You guys see this?
There's one other word of
caution I would like to give you
on the using of this thing.
It's the other word of caution.
I didn't mention it because it's
usually a little bit obvious.
But what's the problem say
applying this thing to something
like 3x plus 1 equals 0.
>> You don't have a -
[inaudible] you don't have a b?
>> There's no a?
>> There's no a.
>> There's no a, yeah.
>> Do you guys see that?
So if there's a equals 0, right,
which is really what's happening
here, what does that cause
when I try to fill out
the Quadratic Formula?
>> You're missing something.
[ Inaudible ]
>> I'm dividing by 0, right?
So this thing works really
when a is not 0 and this.
So really, I should
write here with a not 0.
But that's going to break
in a much more obvious way.
I'm not going to try do divide
by 0, I know that's not allowed.
Cool? Are we good?
