>> This is Part 6 of
completing the square.
In this video, we derive
the quadratic formula
by completing the square.
So we start off with AX
squared plus BX plus C equals 0.
And we solve for X
to get this formula.
And then we learn the quadratic
formula song so it's easy
to remember the formula,
and we solve this equation
using the quadratic formula.
All right.
We're going to try
and find the solutions
of AX squared plus BX
plus C, and we're going
to be doing completing
the square.
So you will have had to have
had some instruction on that
to fully understand this.
All right.
So step 1 is I've got something
in standard form
so it's equal to 0.
It's an equation.
I'm going to start off
by dividing both sides
by the coefficient of X squared,
which is A. So divide everything
by A. Now, notice
I put each term
over A. It's really the same as
putting it all over A. But then,
of course, you'd
break it up -- whoops.
And you could put each
term over A, right?
So we would have AX squared
over A plus BX over A,
which I'm going to write
as B over A times X, okay,
plus C over A. And
what's 0 divided by A?
0. When 0's in the numerator,
you're going to have 0.
And if you remember,
I said A can't be 0,
the coefficient of X squared.
So we don't have to worry
about that we're dividing by 0.
All right.
So this gives me X
squared plus B over AX.
Now, I'm going to take
the constant term C over A
and subtract CA from both sides.
So I'm going to put it on
the other side as negative C
over A. I'm going to
leave a little space here.
It equals negative C over
A. And I'm going to try
to complete the square here.
I'm going to add
something to both sides
of the equation, okay?
Now, what's it going to be?
Well, if this was a
perfect square, then,
to get an X squared
here, it's going to have
to be X times X to
get the X squared.
So that's an X. Since this is a
plus, this middle term's a plus.
And, then, hopefully you
remember, this term in here,
you're going to multiply
it by 2 to get B over A.
So it's actually half of
B over A to begin with.
So it's got to be half of
the coefficient of X. So half
of B over A is B over 2A.
So let's see if that would work.
If I'm going to use my formula
for squaring a binomial,
look down here.
I would get X squared.
So X times X is X squared.
And then I would have,
for the middle term,
remember it's X times
B over A --
I mean X times B
over 2A times 2.
So convince yourself if
you did X -- let's see.
Let's move this up.
And what is X times
B over 2A times 2.
Did you see all the 2s cancel?
And, lo and behold,
I get B over AX.
So that's correct
for the middle term.
And, then, the last term
which is what we want
to know what goes in the box
here is the last term squared.
So B over 2A times B over 2A
is B squared over 4A squared.
Whatever I add to the
left side of the equation,
I must add to the right
side of the equation.
Okay. So I have B squared over
4A squared minus C over A. Ooh,
how are we going to simplify
on the right-hand side?
I have some fractions, so I need
to get a common denominator.
That common denominator
is going to be 4A squared.
So this has to be
multiplied by 4A over 4A.
Let's see.
Okay. Hopefully, you
can see all that.
All right.
So what do we have?
Keep on going.
So I've got this X
plus B over 2A squared,
and we have a common
denominator.
I have B squared over
4A squared minus --
I'm going to write this as 4AC,
just write in alphabetical
order, over 4A squared.
So I have X plus B over 2A
squared is B squared minus 4AC
over 4A squared.
Okay. Now, so I know this
thing's squared is what's
over here.
And now to simplify -- I
mean to figure out what X is,
I'm going to take the
square root of both sides.
Okay. So when you take the
square root of both sides
[ Pause ]
remember you have to
put a plus or minus
in front of one of the sides.
All right.
So if I take the square
root of something squared,
I get what's inside,
which is X plus B over 2A.
And on this side, I have a
plus or minus the square root
of the top, which is
B squared minus 4AC
over the square root
of the bottom.
Well, the square root
of 4A squared is 2A.
Almost done.
So now I'm going to subtract
B over 2A from both sides.
So I get negative B over 2A.
Then I have this plus or minus,
B squared minus 4AC over 2A.
And, great!
Common denominator.
So I have it all over 2A.
We have negative B plus
or minus the square root
of B squared minus
4AC all over 2A.
Awesome! There it is!
The quadratic formula.
This is what we just discovered.
If you have a quadratic equation
in the form AX squared plus
BX plus C equals 0, where A --
that's the coefficient
of X squared -- B --
that's the coefficient of
X -- and C is the constant,
then you don't have to
complete the square anymore.
The solutions for X are
simply negative B plus
or minus the square root of B
squared minus 4AC all over 2A.
This is called the
quadratic formula.
Because now it's a formula,
if you're given an equation
in this form, you can just
plug in the values for A, B,
and C and simplify this and
you'll have the solutions
to X. There's a song we can
sing to learn this and it goes
to "Pop Goes the Weasel."
[Singing] "X equals negative B
plus or minus the square root
of B squared minus
4AC all over 2A."
Now, there's more to it, but
I'm going to leave it for now.
So this is the main one.
So sing it with me.
You ready?
X equals negative B plus
or minus the square root
of B squared minus
4AC all over 2A.
Okay? So listen.
Sing it about ten times
for five days in a row
and you're never going
to forget this song.
So here's a problem.
What's A, B, and C?
Well, A is the coefficient of
X squared, B is the coefficient
of X, and C is negative 7.
I like to do B squared
minus 4AC first.
That's called the discriminate.
So B squared is negative
3 times negative 3.
It's 9 minus 4AC.
Right? I'm going
to do A times C,
2 times negative
7 is negative 14.
So I have 9 plus
56, which is 65.
All right.
So I know where I'm going to
plug in for B squared minus 4AC.
I've already done that part.
So now let's use the
quadratic formula.
It's X equals negative B.
That means the opposite of B,
3 plus or minus the
square root of 65.
I already did the B squared
minus 4AC all over 2A, A is 2.
And there you go.
La di da di da.
Two solutions: 3 plus the
square root of 65 all over 4
and 3 minus the square
root of 65 over 4.
These are really two solutions.
Okay. So we've got
the quadratic formula,
how I derived the quadratic
formula, how to sing it
so you can memorize it, and
just one little example.
Have fun.
[ Silence ]
