Hi there. So, I'm gonna start 
chapter three now. 
Um, chapter three is on 
quadratic functions. 
However, before we actually 
get into the meat of the section, 
in the, um, at the 
end of the chapter,
uh, there is something 
called a skills refresher, 
and that's what I would like 
to go through, um, today.
'Kay? So. This is a skills refresher, 
I'm gonna call it section 3.0...
but we're gonna use this 
as a skills refresher. 
Now, one other quick thing 
that I would like to say 
is that I'm assuming that you guys have 
done these things before, in the past, 
I'm just trying to, uh, 
refresh you guys. 'Kay?
One of the things we're gonna 
be talking about is factoring 
and then the other thing we're gonna be 
talking about is called completing the square.
Now, the important thing before 
we actually start factoring 
is to be sure that you know how 
to use the distributive property, okay? 
So, what's an example? 
Well, using the distributive property, 
suppose I have 
negative 4 times 
the quantity y plus 6. 
Well, what this says is distribute 
your 4 throughout your parentheses. 
So I claim we get 
negative 4y minus 24.
'Kay? And we could 
do another one, 
suppose I have say 3z times 
the quantity 2x minus 9z.
This might not be very pretty, 
but we can do this. 
'Kay? So, again, you just distribute it 
throughout your parentheses,
so if I take 3z times 2x, 
I claim I get 6xz,
now yes, 
you could put zx, 
I just like to put things 
in alphabetical order if I can.
Minus, well 3 times 9 is 27, 
and z times z is z squared.
Okay.
Now let's do another one. 
What if I have x times 
the quantity 3x minus 8,
plus 2 times 
the quantity 3x minus 8.
Alright. Now, 
instead of distributing, 
what I want us to do is essentially 
combine like terms and factor.
'Kay? I mean, one thing 
we could do is we could say,
well this is 3x squared 
minus 8x plus 6x minus 16.
'Kay? And then 
combine like terms.
'Kay? But what if--
yeah, what if I wanted to say 
instead of using the distributive property,
what if we wanted to 
actually factor this guy.
'Kay? This is kind of jumping a little bit ahead 
but I think it's really important.
'Kay? Notice--one of the things that you do 
when you factor is you look for common terms,
these guys both have 
a 3x minus 8 in common.
So I claim we can 
pull out 3x minus 8.
'Kay? I'm doing kind of 
the anti-distributive property. 
If I pull this 3x minus 8 
out of of the first term, 
well, he's gone,
and x is left,
and if I pull the 3x minus 8 
out of my second term, 
he's gone, and I'm just left
with an x plus 2.
I know that's kind of funky, but it's really important 
to be able to look for things like that.
Okay.
Now what I would like to do is
practice with one foiling guy.
Let's do this: uh, suppose 
I want you...to multiply this out.
'Kay? This is where we use 
that acronym, FOIL?
First, outside, inside, last.
'Kay? If we multiply 
our first terms we get 10--
that did not show up very well. 
10x squared.
Our outside terms,
we get minus 15x,
our inside terms,
minus 2x, 
and then our last term's 
plus three. 
So then if we combine like terms 
we get 10x squared minus 17x plus 3.
Okay. 
Now. What I would like to do is talk about 
how in general when you need to factor something
what are the steps that you 
should go through in order to do such.
'Kay? So. 
When you factor.
The first thing that you should do
no matter what you're factoring, 
is pull out any
common factors.
For example, what if I have 
4x minus 6 and I wanna factor that.
'Kay? Well, remember terms are things 
that are separated by plus and minus signs,
what do both 4 and 6 
have in common? 
I claim 2 goes 
into each of them.
Well, if I divide 
my first term by 2, 
I claim I'm left with a 2x,
and if I divide my 
second term by 2,
I claim I have 
a minus 3.
Now the beautiful thing about this 
is you can just check your answer 
by using the distributive property: 
4x minus 6.
Alright, what if we 
wanna do this guy?
Okay, how about, 
um, 3y plus 3.
What do these guys 
have in common? 
Well, I claim there's 
a three in each term.
If I divide my first term by three,
I have a y left.
Now, I will grant you 
that if I pull the three out,
um, it looks like on first sight, 
oh, there's nothing there. 
But remember your dividing by 3 
and 3 divided by 3 is 1.
Okay. What if I have 
something like this?
3u to the 4th,
minus 4u cubed.
Now in terms of our numbers, 
there is nothing in common.
However, here 
I have four u's, 
and here I only 
have three u's.
So, what can 
we pull out, 
I claim you only can pull out 
the smaller amount.
So what do I have left in my first term, 
well I still have this 3, 
I started with 4 u's, 
I took three out so I have 1 left.
And in this next term, 
notice I took out all my u's, 
but I still do have 
that minus 4.
Okay. So. No matter 
what you're doing, 
when you factor, the first thing you gotta do 
is pull out any common factors.
The next thing that 
you should do is as follows:
you should note that if 
what you're factoring has 2 terms,
you should be on the lookout for what we call 
the difference of 2 squares.
And this is a formula that you 
should have seen before. 
'Kay?
Essentially what this says is 
if your first term is squared  
and you're subtracting off 
another term that's squared. 
That's why it's called difference, 
because we're subtracting.
'Kay? I claim what you do is you take 
the first guy that's being squared,
add it to the second guy 
that's being squared,
times the first guy 
that's being squared
minus the second guy 
that's being squared.
If you don't believe me that it factors this way,
well, we can verify that, right?
We can foil this out and then we 
should get what we have over here?
Let's do that.
x times x is x squared,
our outside terms 
give us minus xy,
the inside terms 
give us plus xy,
and the last terms give us 
minus y squared.
What happens to 
these two terms? 
They cancel each other out,
and lo and behold, we've got it.
So. For example, what if 
I have x squared minus 9?
Well, would you agree that that's 
x squared minus 3 squared?
So, to factor that, you take 
the first thing that's being squared
and add it to the second thing 
that's being squared,
and then the first thing that's being squared
minus the second thing that's being squared.
And again, the beauty of this stuff 
is you can always check your answers.
Okay.
Next. 
Oh, let me just do one more example, 
um, while I'm thinking of it, 
why don't we do...hmm,
oh, this is cute. 
9x  squared 
minus 9.
'Kay? Suppose I want us 
to factor that.
Well, what's the first thing you should do 
whenever you factor something? 
Pull out any 
common factors.
So. Let's look 
right here. 
What's in common in both of my terms?
We have a 9.
So if I pull a 9 out of my first term, 
I'm left with an x squared.
If I pull a 9 out of my second term,
I'm left with minus 1. 
But I claim 
I'm not finished.
Why? Because, I claim this is actually 
the difference of two squares.
In other words, 
we could say, oh! That's 9,
and then this is take the first thing 
that's being squared 
plus the second thing 
that's being squared,
and then you take it 
times x minus 1. 
Isn't that cute?
Alright.
So. The fourth step with...or, excuse me, 
the third step with factoring, 
you'll see why I said fourth 
in a second--
[laughs]
is--if there are 
four terms,
you wanna factor
by grouping.
And I claim that's like doing 
step one two times. 'Kay?
Let me show you 
an example:
x cubed minus 2x squared
plus 3x-6.
'Kay? Now. What's the first thing 
that we should do 
whenever we approach 
a factoring question? 
We should pull out 
any common factors.
But I claim we have nothing 
we can pull out of here.
Okay? So, there are two terms,
nothing to do there.
But there are four terms. 
So how do you factor by grouping? 
Essentially what you do is you group 
your first two terms and your second two terms.
And then you pull out 
any common factors. 'Kay?
What do these guys 
have in common? 
I claim we can pull out 
an x squared.
If I do, I'm left with 
an x minus 2.
In my second group, 
I claim I can pull out a 3.
If I do, I'm left 
with an x minus 2.
Now this should ring back to an example 
I did just a moment ago:
notice now we can still 
pull out a common factor, 
because both of these terms 
have that x minus 2.
If I pull out the x minus 2 
from my first term,
I'm left with this 
x squared,
and if I pull the x minus 2 
out of my second term,
I'm left with 
plus 3.
Isn't that cool?
'Kay? Let's do 
another one. 
Alright, what if now I have x cubed plus 
4x squared minus 3x minus 12?
Alright. Again, 
group our terms,
I can pull an x squared 
out of my first group,
I claim I'm left 
with x plus 4,
and over here,
I want to look ahead and say 
gosh, I want to pull something out so that 
I can be left with a positive x and a positive 4. 
Notice both of these 
are negative.
So I'm gonna pull out 
a negative 3.
If I take this negative 3x divided by a negative 3, 
I have a positive x.
Negative 12 over negative 3
is a positive 4.
Again, I've got this 
term in common, 
I pull out 
my x plus 4,
if I do that I'm left with 
an x squared in my first term,
and a minus 3 
in my second term.
Am I done? Actually I am, 'cause notice 
three is not a perfect square. 
Alright. One more example. 
For now.
What if I have x cubed 
minus x squared minus 9x plus 9.
Alright. Again,
let's factor by grouping, 
I can pull an x squared 
out of my first term,
well, I'm left with 
an x minus 1.
Now, notice my first term here is negative, 
my second one is positive,
but I'd like to pull something out 
so I'm left with a positive x. 
So what I'm gonna do 
is pull out a negative 9.
If I pull out a negative 9, 
I'm left with a positive x,
and then notice 9 over negative 9
is a negative 1.
'Kay? Next, I can 
pull out my x minus 1,
if I do that I'm left with 
an x squared in my first term,
and a -9 in my second term.
I claim this time 
I'm not done.
Why? Because 9 
is a perfect square. 
Isn't this the same as 
x squared minus 3 squared?
So? Don't forget to rewrite this guy, 
we have an x minus 1,
and then we factor that 
to be x plus 3, x minus 3.
I like that.
Okay. 
Finally, step four, is 
if there are three terms.
'Kay? The first thing that 
we're going to be dealing with 
is if we have a coefficient of 1 
in front of our x squared term.
Cause that's a lot easier 
of a situation. 
For example, what if we wanna factor 
x squared plus 5x plus 6.
'Kay? Notice there's a coefficient of 1 
in front of my squared term.
When you approach 
these guys, 
well, you're kinda 
anti-foiling if you will.
'Kay? My first terms I know 
are each gonna be an x, 
x times x gives me 
x squared.
Now, the way I wanna look at this 
is I wanna find two numbers
whose product is 
the last number
and whose sum 
is the middle number.
'Kay? Well if I wanna multiply up to a positive 
and add up to a positive, 
I claim both of them 
have to be positive.
So what two numbers multiply up to 6 
and add up to 5?
3 and 2. 
'Kay? 
Now let's do this guy. 
I'm gonna keep using these numbers 
or something very close. 
What if now I wanted to do 
x squared plus 5x minus 6?
Well, again, I--
we'll have x and x here 
but now I want multiply 
up to a negative number. 
The only way you can multiply 
up to a negative number 
is if one of them is positive 
and one of them is negative.
Since my sum 
is positive, 
I know that my bigger number 
has to be positive.
So now 3 and 2 won't work but I claim 
positive 6 and negative 1 will work.
'Kay?
On the other hand, what if I have 
x squared minus 5x minus 6?
Again, they're gonna 
start out with x's,
and again one is positive 
and one is negative 
cause that's the only way we can 
multiply up to a negative number.
But now what I want is them 
to add up to a negative five. 
Well, since the sum is negative, 
I know my larger number is negative 
and I claim negative 6 
and positive 1 work.
'Kay? Two more 
along this line.
Let's do x squared
minus x minus 6.
'Kay? Well, I know 
we got x's at the start,
I know that they have to multiply up 
to a negative number, 
so I know one is positive 
and one is negative 
but now notice they need 
to add to a negative 1.
Again, since the sum is negative, 
the larger one is negative, 
and in this case a positive 2 
and a negative 3 work.
Alright?
Now, what if the number in front 
of your x squared is not a 1?
'Kay? 
Then these can be 
a little more of a pain, 
but they're certainly doable.
So what if I have something like this.
Suppose I have this guy: 
6x squared
plus plus 6x--
one second. 
Uhhh, 6x squared...
oh, excuse me. Plus 7x, I was gonna say, 
that's not gonna work, 
minus 3.
'Kay? Now. What I wanna present to you 
are all the different issues we have going on, 
because this isn't a 1 
in front of here.
'Kay? We can do this 
by trial and error.
Do you agree that the only way I can multiply 
my first terms to get a 6x squared 
is if I have 6x 
and an x?
Or a 2x and a 3x?
'Kay? So already we're starting 
to introduce a lot of possibilities. 
Now, there aren't a whole 
heck of a lot here,
because my last number's 
a prime number, 
it's a 3 and the only way you can multiply up 
and get 3 is if you have a 1 and a 3,
but notice one is positive, 
one is negative. 
So what I'm gonna do is I'm gonna write 
some general steps on how you would solve this.
'Kay? So, when you don't have a 1 
in front of your x squared term, 
what I would recommend you do is
you need to find two numbers...
whose product is the first number
times the last number.
And whose sum is 
the middle number.
'Kay? In other words, we wanna find two numbers whose product is negative 18
and whose sum
is seven.
Now if it helps to write out all the things 
that multiply up to 18,
go for it, 
'kay? 
Notice the product is a negative number 
and the sum is a positive number 
so we know one is positive 
and one is negative.
But since my sum is positive we know 
that the bigger number is positive.
I think you'll grant me that 
9 and negative 2 work.
'Kay?
So. We found 
those numbers. 
By the way, if you can't find those numbers,
this isn't factorable.
Step two: what we do then 
is rewrite the middle term...
using those 
numbers.
Why? Because step three
will be factor by grouping.
Let me show you 
how this works.
'Kay. So we found 
these numbers. 
Now step two says rewrite 
the middle term using them. 
In other words, 
we have 6x squared,
but in place of 7x, I'm gonna say well, 
that's a 9x minus 2x. 
'Kay? I haven't changed the value of anything, 
all I did is rewrite the middle term, 
but I claim that if you combine like terms,
what we have is equivalent to what we have above.
Now notice:
what do we have? 
We've got something
with four terms. 
We can factor him 
by grouping.
What can we pull out 
of our first terms?
I claim we can pull out a 3x, and if I do that, 
well, I'm left with a 2x plus 3.
Now I ask myself, is there anything 
I can pull out of my second group.
Offhandedly, no.
'Kay? 
2 and 3 don't have 
anybody in common. 
But I do claim I can 
pull out a negative 1,
and then I'm left with 
a positive 2x and a positive 3.
And now these guys have 
a 2x plus 3 in common,
I can pull 'em out and 
I'm left with a 3x minus 1.
Alright. Let's do 
one more example there 
and then we're gonna keep playing, 
don't you worry.
But let me pick 
a good example.
So now, what if I had 
6x squared plus x minus 12.
Now again, if you wanna do these 
by trial and error, 
just playing around 
with different possibilities, 
you're more than welcome to, 
but notice: 
we could either start with 
a 2x3x or and x and a 6x, 
but there's a lot of different ways 
we can multiply up to a 12.
So let's use the technique 
we've established.
We wanna find two numbers 
whose product
is our first number times our last number 
which is negative 72,
and whose sum is 1.
Well, play around with 
all the factors of 72, 
and I claim a positive 9 and the -8 
are gonna give it to us.
These guys multiply up to 72 
and they add up to a positive 1.
So, next step: rewrite our middle term
using those numbers.
So we have 6x squared plus 9x
minus 8x minus 12.
And now we'll factor 
by grouping.
I can pull a 3x 
out of my first group,
and I think we've seen this before. 
Look right over there. 
We're left with 
the 2x plus 3.
'Kay? What can I pull out 
of my second group?
Well, I can pull out a negative 4
and I'm left with a 2x plus 3.
I pull out my 2x plus 3,
and I'm left with a 3x in my first term
and a negative 4 in my second term.
So I hope this helped in terms of, 
um, factoring. 
Now. One of the things 
that we tend to do, 
and you'll see why soon enough, 
once we get into the meat of chapter three,
is we want to solve 
various quadratics. 
And one way to solve them 
is by factoring.
But the key thing to remember 
is we have our property of zero.
What that says is if you multiply
two things together and get 0,
so if a times b 
is equal to 0,
then what we can conclude is either 
a is equal to zero or b is equal to zero or both.
Right? The only way you can 
multiply things together to get zero 
is if one or both 
are equal to zero.
'Kay? So. 
For example. 
What if I wanted us to solve x cubed minus 
2x squared minus 3x equal to 0.
And I'm gonna tell you we're gonna do this 
by factoring.
What's the first thing you do whenever you factor?
You pull out what's in common.
What can I oull out? 
I can pull out an x.
'Kay? If I pull one x out of here 
I still have two left.
If I pull an x out of my second term,
I'm left with minus 2x, and then minus 3.
Next step, 
let's factor this guy.
You still have to write this first x, 
he still carries through.
Notice these are both 
gonna start with an x, 
and since I'm multiplying up 
to a negative number 
I certainly know that
one is positive and one is negative.
So.This is nice, 
we don't have a lot of choice. 
What numbers multiply 
up to 3? 1 and 3.
But since the sum is negative, 
I know my larger one is negative.
Alright then. We factored it, 
now notice:
yes, we have three terms here, 
but we can extend this. 
The only way you can multiply 
things together to get 0
is if  the first one is 0, the second one is 0,
or the third one is 0.
In other words, if x is equal to zero,
if x is equal to negative one, or if x is equal to 3.
Isn't that fun? 
'Kay. One more.
Okay,  what if 
I have this: 
this is just 
a factor.
x squared e to the negative 3x 
plus 2xe to the negative 3x.
'Kay? Don't get messed up 
by this e business, 'kay? 
We will be talking about our 
exponential functions with base e
later on 
in this class, 
but right now the way you want to approach this 
is you wanna pull out whatever is in common.
Okay? 
Now.
What do we have? We have an x squared here 
and an x here. 
So we can pull out 
one of those x's. 
And don't I have an e to the negative 3x 
in both terms?
If I pull that out, 
notice:
this is gone, I had two x's, 
I pulled one out, so I've got 1x left,
and over here I pulled all this out 
and I'm just left with plus 2.
'Kay? So, if you see a homework problem 
with these e's right now
don't let that freak you out.
Mmkay? You will understand the meaning of it 
soon enough.
But I just wanted to give you 
a heads up.
Alright. So. 
That's factoring. 
Now what we need to do is 
talk about completing the square.
'Kay? We like 
perfect squares. 
1, 4, 9, 16, 
25, etc., 
because they're actually equal 
to some quantity squared.
But we can also look at our quadratics 
as perfect squares.
For example, x plus 2, 
quantity squared. 
If you multiply that out, trust me, 
that's equal to x squared plus 4x plus 4.
'Kay? So we say, oh, this could be written 
as some quanity squared. 
It's x plus 2 times x plus 2, 
aka x plus 2 squared.
I claim another perfect square is something like, 
how 'bout x squared minus 6x plus 9?
'Kay? I claim, and you can verify,
that that's equal to x minus 3 quantity squared.
If you multiplied out x minus 3 
times x minus 3,
trust me. You're gonna get back 
to where we're going.
'Kay? So. 
One of the things that we like to do is 
be able to write things in terms of perfect squares.
'Kay? So. 
How do we do that? 
Well, I'm gonna write out 
some things in general, 
and then what we're gonna do is apply it 
into a bunch of different specific examples.
'Kay? The rules for 
completing the square
is first off you need a coefficient of 1
in front of your squared term. 
'Kay? If need be, 
you can factor him out.
Two, and again these steps 
are gonna seem funky, bear with me.
What you do is you take 
half of the middle term...
and you write it 
to the side.
'Kay?
Step three.
You square that number
and then what you do is you add,
I'm gonna say and compensate.
That's gonna seem strange,
it's okay. 
The key in life with anything is balance
and that carries over in mathematics.
Okay?
So.
Let's go through 
and start these. 
We're gonna 
build slowly
and you'll see how this is gonna relate
in a little bit in this section. 
So for example, if we want to complete the square for this guy, x squared plus 8x, 'kay? 
I know every part of you 
wants to factor out next, 
and I get it, but that's not 
what we're doing here. 
We want to complete 
the square here.
Well, notice we have a coefficient of 1 
in front of our squared term, life is good.
Now.
What we want to do, normally there's a third term, 
which is why I call this the middle term,
take half of this number.
You get a 4. 'Kay?
Now what we have to do 
is square it.
'Kay? And if you square 4, 
I claim you get 16.
'Kay?
Now why would we 
want to do this? 
This is just a 
prelude example. 
Because I claim you can then write it 
as a perfect square.
What number 
should go here?
Whatever you got when 
you took half of that middle number.
'Kay? So. How would we end up
using this? 
Well, let me show you
an example here. 
What if we have this: 
x squared plus 4x minus 5.
'Kay? Now we actually 
have a middle term.
[coughs] 'Scuse me. 
And we want to complete 
the square here. 
'Kay? 
Now.
Our goal in the end 
is to write it in this form.
Some number...
like this.
You'll see why,
just hold on.
'Kay? So: what we do 
is we take,
notice you have 
a coefficient of one here, 
you take half your middle term.
Half of 4 is 2.
And what do we do with that? 
We square it. 
So, x squared 
plus 4x plus 4.
Now, I still have 
this minus 5.
'Kay? I've completed 
my square here. 
I'm gonna get to that compensate or balance comment that I made over there.
'Kay? Can you just add 4 onto an equation 
and think that you've got the same equation?
Absolutely not.
'Kay?
So I need to balance it. If I add 4 here 
in order to keep the same thing,
I've got to subtract 
4 out there.
'Kay? Now what we'll do is 
write this as a perfect square, 
what number 
goes here?
Well, that 2 that we wrote 
on the side.
'Kay? Alright, 
let's do this one.
What if I have...
mmm...oh, this is a good one. 
Umm, negative 4x squared, 
plus 8x minus 3.
Notice I do not have a coefficient of 1
in front of my squared term.
'Kay? When you don't, what you're going to do
is the following:
factor it out of 
the first two terms only.
'Kay? So: I'm gonna pull that 
negative 4 out, 
and I'm left then with 
x squared minus 2x.
Notice I only pulled him 
out of my first two terms.
'Kay? Again, you can distribute and verify 
that you get the exact same thing. 
We're not changing 
our equation at all.
Okay. Now what we're going to do 
is complete the square. 
Things are gonna get 
a little funnier here though.
Ready?
'Kay? 
I'm gonna complete 
the square with this term.
Well, how do you do it, you take half of this number. What's half of negative 2?
I claim negative 1.
I always write 'em 
off to the side.
When I square that negative 1, 
I claim I get a plus 1.
'Kay? I still have 
that minus 3,
but notice: I've just added 
an extra term here.
'Kay? That's where 
this balance comes in.
'Kay? And yeah.  
In parentheses 
I added a 1. 
But notice, there is that 
negative 4 out there, 
so haven't I essentially subtracted 4
from my equation if I was gonna distribute that?
And if I subtract 4
in order to keep balance, 
I have to add it on 
at the end.
So, now I wanna rewrite this 
as something squared,
again, what number 
goes here?
What we got when we took 
half that middle term or negative 1.
And that's what we have.
'Kay?
Let's do another one of these.
What if I want to 
do this one?
Mmm, let's actually 
change it slightly, 
what if now instead of solving by factoring, 
we wanna solve by completing the square.
What if I have 
x squared...
let's do this: 3x squared
minus 6x minus 7.
Alright. How would we go through 
and solve that.
Let's actually make this a little nicer for ourselves 
and start out with an x squared.
First off, why did I change that? 
'Cause I wanna point something out.
Couldn't we factor this?
Yeah! 'Kay? As an aside, I know, 
oh, sweet! That's x minus 7 x plus 1.
Right?
So.
Got the product of 
two things equal to 0. 
Either the first one is 0 
or the second one is 0.
Doesn't that give us x equals 7 
or x equals negative 1?
'Kay? Let's do this by 
completing the square, though.
'Kay? This is cool.
Ready?
If you're going to solve something 
by completing the square, 
the first thing you do is 
move your constant to the other side.
So we have x squared 
minus 6x equals 7.
'Kay? Next: 
complete the square.
How do you do it? You take half of this middle term,
half of negative 6 is negative 3.
What do you do 
with that negative 3?
You square 'em.
Now. Instead of adding, subtracting to balance, 
now I have an equation. 
So if I'm gonna add a number to one side 
of my equation, don't I add it to the other side?
'Kay? Next step, I think you'll grant me 
9 and 7 is 16,
what number 
goes here?
The number we wrote off to the side 
when we take have of that middle term.
Alright. How would 
we solve this? 
Well, I'm certain you've seen this before 
at some point in your life, 
you use the square root method.
You got some quantity squared, 
so we can take the square root of both sides. 
On our left side 
we get an x minus 3,
and we're in luck, on our right side, 
we can get plus or minus 4. 
Again, you don't just consider 
the principle square root, 
you consider both your positive 
and your negative values.
So. In order to get x all alone, 
I can add 3 to both sides,
and don't I have x then 
is 3 plus or minus 4?
In other words, one solution 
is 3 plus 4, a.k.a. 7, 
and the other solution is 3 minus 4,
a.k.a.  negative 1. 
Notice you get the same thing
either way.
That is so cool.
Alright. Now again, I'm just going through 
a few examples here, 
because you guys have probably seen this 
at some point in your life,
at least that's my hope.
Now, perhaps we should do 
one more example,
and then I'm gonna show you 
something so beautiful.
I'm really excited about it.
Okay?
Before I show you my 
such a beautiful thing, 
let's do one more example of
completing the square and solving.
What if I have this:
how about y equals...
oh, no, no, no, no, no. 
Let's do this.
We're gonna solve. 
So how about we have, um, 2x squared 
plus 4x minus 5 equal to 0.
'Kay? If I asked you to factor this, 
you wouldn't be able to, 'kay? 
He's not a factorable one.
So. What's my 
first step?
Well notice, I've got a coefficient of a 2 
in front of my squared term. 
'Kay? So. 
When you're solving, 
essentially what I would recommend you do 
is just divide everybody by that coefficient.
'Kay? If I do that, I have an x squared plus 2x minus 5 halves equals 0. 
Are we kind of bummed that we got a fraction? Yeah, I am. But it's okay.
Alright? So. Now I have a coefficient of 1 
in front of my squared term.
'Kay? So. 
I do wanna point out
that when you're just, um, when you're not solving something equal to 0, 
if I just have y equals this,
then I would just factor 
the 2 out of my first terms,
and go through 
and complete the square. 
But when you're actually solving,
you need to divide all the terms by him.
'Kay? It's an important 
distinction to make.
'Kay, my next step is move my constant 
to the other side. 
I added him, now what I'm gonna do 
is complete the square. 
I take half of this middle number,
half of a positive 2 is a positive 1,
I square him 
and I get 1.
Now, fair is fair, if I add it to one side, 
I'm gonna add him to the other side.
I can write this side as a perfect square,
what number goes here?
What I got when I took 
half of my middle term.
And 5 halves plus essentially 2 halves 
is 7 halves.
Now, in order to solve this, what I'm gonna do 
is take the square root of both sides.
I didn't say 
it's pretty.
Alright, the square root 
of something squared is itself,
and here I'm gonna get positive and negative 
the square root of 7 halves.
'Kay, so if we were leaving this 
as an exact answer,
well, we get x alone 
by subtracting 1, 
so we could say negative 1 plus or minus 
the square root of 7 halves.
And for our purposes right now 
I'm just gonna leave it like this.
Again, one solution is negative 1 
plus the square root of 7 halves,
and the other solution is negative 1
minus the square root of 7 halves.
'Kay?
Now I'm gonna show you something that's really 
cute, cool, beautiful, all of those things combined.
'Kay? I would not ask you 
to do this on your own,
not that it's dangerous 
for your health,
but this is so cool.
Ready? I wanna complete the square
for this.
'Kay?
Now. My only rule, 
my only stipulation, 
is a can't be 
equal to 0. 
'Kay?
You might ask why would I want to solve something like this by completing the square?
Hold on a second 
and you will see. 'Kay?
So. Remember, it we're actually solving 
this equal to 0 in order to complete the square 
the way we get a coefficient of 1
in front of our first term 
is we divide everybody by that.
Yes, this is kinda 
yucky looking, 
it's okay, cause 
I'm the one doing it anyway.
So. If I divide through by a 
I have x squared,
plus b over ax plus c 
over a equal to zero.
'Kay?
Again, a's not 0 so it was totally legal 
to divide by him.
Our next step? 
Move our constant 
to the other side. 
I have x squared plus b over ax
equals negative c over a. 
Alright.
Ready?
Next, we want to 
complete the square.
How do you complete the square? 
You take half of your middle term.
Remember in mathematics, 
the word 'of' means multiply. 
So if I'm taking one half
of b over a,
I'm multiplying them.
When you multiply fractions 
you multiply your numerators 
and you multiply your denominators.
So when I take half of my middle term, 
I got b over 2a.
The thing is, next step when 
you're completing the square 
is you wanna 
square him.
'Kay? Well, what do you get 
when you square b over 2a, 
well you get b squared,
2 squared is 4, 8 squared.
So, I'm gonna add on b squared 
over 4a squared to both sides. 
b squared over 4a squared
and then we have that.
I know this looks nasty 
but just hold on.
You're gonna have a peak experience here, 
seriously.
Okay.
So.
I claim even though our left side looks terrible, 
it's actually the easier side to deal with, 
because he's 
a perfect square.
Again, what number 
goes here?
Well, what we got when we took 
half of our middle term, b over 2a.
Now on our right side, 
we can't combine those yet. 
We need a common denominator.
'Kay? I have a 4a squared 
here and an a here,
so my common denominator I claim 
would be 4a squared.
How do you get it?
Well, don't we need another 
4a down here?
And again, fractions 
are like kids: 
whatever you do to your denominator 
you gotta do to your numerator.
So I claim we have b squared over 4a squared
minus 4ac over 4a squared.
Well, we got a common denominator,
so don't we have b squared minus 4ac
in our numerator?
Our next step is to take the square root 
of both sides,
the square root of some quantity squared
is itself,
we get plus or minus,
well, we can break this big square root up 
to be the square root of our numerator
over the square root 
of our denominator.
But let me ask you this: 
what's the square root of 4a squared? 
Well, the square root of 4 is 2,
the square root of a squared is a.
So, by moving this over, notice I have 
a common denominator of 2a,
this would give me a negative b, 
plus or minus that square root. 
And look at what we have.
When you're solving a general quadratic set 
equal to 0,
here's our solution.
I bet you've seen this before, 
this is our quadratic formula.
That is so cool. So if you've ever wondered where 
your quadratic formula came from, 
it came precisely from 
completing the square.
'Kay? Again. That's not something 
I'm going to ask you to do on your own,
but I think it's cool to see where 
a formula actually comes from, 
because it actually 
gives meaning to it.
Okay? So. Just to summarize, and then I will solve 
one example using my quadratic formula,
and then we'll call this lesson done,
is this:
factoring is 
really fun.
'Kay? The thing is, 
is it works rarely.
So, the bummer with factoring is 
it doesn't always work.
'Kay? You can also solve these guys 
by complete--completing the square
or by using your 
quadratic formula.
These two guys
always work.
'Kay? Now, honestly, if I had my druthers
and I wanted to solve a quadratic, 
I would use my quadratic formula 
over completing the square.
You'll see the utility of completing the square
in chapter three and how that's relevant, 
however if I'm actually solving a quadratic
and I noticed that it can't be factored, 
my own personal thing is always 
just use my quadratic formula.
So let's do one example using our 
quadratic formula, and that's that. 
So 2x squared plus 6x 
minus 3 equal to 0.
Your quadratic formula, 
I'll rewrite it here, 
says when you have this, 
the solution is negative b,
plus or minus the square root of b 
squared minus 4ac,
all over 2a.
a is the number in front of x squared, 
b is the number in front of x, 
and c is your constant term.
'Kay. Let's do this. 
So. First thing, when you're 
just starting this out, 
it might be useful to identify 
what's a, b and c.
a is 2, b is 6,
and c is negative 3. 
Let's plug it in our formula.
Negative b. 
Both b is a positive six, 
negative b is negative 6, 
plus or minus the square root 
of b squared,
minus 4,
times a, times c
all over
2 times a.
'Kay. Let's see 
what we get: 
negative 6 
plus or minus. 
Well, 6 squared is 36, 
4 times 2 is 8, times 3 is 24,
but notice minus a negative 
is plus a positive,
and that whole thing 
is over 4.
Alright.
So you get negative 6 
plus or minus, 
well, what's 36 
plus 24,
that's 60
over 4.
Now. This can be 
simplified.
Remember the way you can simplify it 
is to simplify a square root?
Yes, you can get a decimal approximation 
but I like to leave things in exact form,
I claim, well, 
can we simplify 60.
Well, how would 
we do that? 
What you would do essentially is 
say how can we break up the number 60
so at least we can take 
the square root of part of it.
And I claim 60 
is 4 times 15.
So, you get negative 6 
plus or minus, 
the square root of 4 is 2,
root 15 over 4. 
Right? You couldn't take 
the square root of all that 
so we broke it up so that we could take 
the square root of at least part of it.
The square root of 4 is 2, and we're left with 
the square root of 15 there.
We tend to like to leave things 
as exact answers.
'Kay? 
Well, one more step.
'Kay?  Notice I've got a 4 here, 
a 6 here and a 2 here.
I claim I can factor 
my numerator.
I can pull out a 2.
And if I do that, I'm left with negative 3
plus or minus the square root of 15 over 4.
Why did I do that? 
Because I could simplify that.
So, I get negative 3 plus or minus 
the square root of 15 over 2. 
I understand it's not necessarily 
the most satisfying answer,
but it is an answer and again 
you can put this into your calculator
and get a decimal 
approximation.
So.
Again.
I'm sure you've seen the quadratic formula before 
and factoring before 
but I urge you to practice 
a ton of these. 
It will only help you.
Thank you.
