Hello and welcome to MySecretMathTutor.
For this video I had a special request to
see how the quadratic formula was built using
the method of completing the square.
Now of course if you are not familiar with
completing the square, go ahead and check
out my video on how that works. Its a really
great way for solving some quadratic equations,
and its of course how the quadratic formula
gets built. Alright.
Let's go ahead and jump right in.
The idea is we are trying to solve a quadratic
equation. We are going to do this in the most
general way possible. So I'm just leaving
a, b, and c as my generic coefficients in
here, and of course those could be any number.
Our goal is to really try and isolate x on
one side, through the method of completing
the square, and of course by the time we are
done, we will have the quadratic formula.
Alright, so, where do we start this process?
Well if you are going to complete the square,
the first thing you want to do is isolate
your variables here. So have this c, I really
want it on the other side, so I'm going to
subtract c from both sides. So that will give
me ax^2 + bx = -c. Ok.
Good place to start. Now I also like to isolate
the x^2 term, make sure that it doesn't have
any coefficients out front, so what I'm going
to do is I'm going to factor it out. And I'm
going to factor it from both these terms.
So I'll have a multiplied by the quantity
x^2 + b/a x and that's all still equal to
-c. Now when you see a step like that its
kinda intuitive, like yea the a is going to
come out, I can see where that came from,
but why all of a sudden does an a show up
over here. It really should end up as a fraction,
and one good way to double check that this
is proper way to factor it out is imagine
taking this a and putting it back in. So a
times x^2, there is my ax^2. If the a multiplies
over here, it would actually cancel with that
a in the bottom, and sure enough we would
just be left with b. So that is the correct
way to factor out that part. Alright, things
are looking pretty good. I want to move that
a to the other side, so let's go ahead and
just divide both sides by a. See where that
takes us.
Alright, so I have x^2 + b/a multiplied by
x equals -c/a.
And I'm really setting this thing us so I
can complete the square. I've made sure my
variables are one side, there is no coefficients
in front of that x^2, in fact I'm in really
good shape that I can actually more forward
and say, "let's go ahead and complete that
square."
Now again this is a really good time if you
haven't seen this method to go back and check
out my other video. Otherwise let's go ahead
and move forward with this guy.
When it comes to completing the square we're
going to look at this term next to x, in this
case the b/a, and we are going to do something
very special to it. We want to divide this
by 2. Let's go ahead and put the little 2
in the basement. Divide by 2. And we are going
to square it. This will of course give us
a brand new number. This will give us b^2
all over, let's see, square the top and bottom,
4a^2. And this new number right here is what
we can use to go ahead and complete the square.
We are going to add this on both sides of
our equation. It looks a little complicated,
looks a little ugly, but it is exactly what
we need to make this thing work out.
Ok, so there is the left side of my equation,
let's go ahead and throw that in there. Plus
b^2 all over 4a^2. And here is the other side
of our equation, of course we have to balance
this equation out. So let's go ahead and add
b^2 all over 4a^2 to the other side.
Alright, so we've put this on both sides,
how does this really help us? Well this is
gong to make the left side factor nicely.
And it does get a little bit tricky when you
get some weird terms in there like b^2 all
over 4a^2, but you know. Let's just focus
on that left side, see what happens, make
sure that it really does factor like it should.
So I'm just going to copy down the right side,
we won't touch that for a bit.
And the way I like to do this is let's just
look at the first terms, what two things would
multiply and give us an x^2. Well those would
both have to be an x. What two things would
have to multiply and give us a b^2, well that
would have to be b, so let's put that in the
top of the little fraction up there, b. And
what would have to multiply by in order to
get a 4a^2, well a 2a multiplied by a 2a,
that would do it. So sure enough I'm setting
this up pretty good, and we have it factored. Now
we have these factions here, again it looks
a little weird, how would my outside and inside
terms really multiply to give me a b/a. It
really does, in fact if we just want to check
it, the outside terms would be x b/2a, plus
x b/2a. They have a common denominator. So
we'd end up with 2xb all over 2a, the 2's
cancel. And there is our x b over a, that
really is how this thing does factor. Alright.
So what does this give us. It says that we
have two factors x + b/2a. And since we have
two, they are being multiplied, I'll say they
are squared. And that is all equal to the
right side. So we've completed the square,
and the really neat part about doing that
is we started with an x^2 and an x, and now
we just have a single x here on the left side.
So now as soon as we do some combining, we get
that x all by itself, sure enough we will
get that quadratic formula. Let's go ahead
and move forward. So the next step in this
is to really try combine as much stuff as
possible here on the right, and then we'll go ahead
and take the square root to move that 2 out
of the way. And if we are going to combine
things over here on the right, we need a common
denominator. Now this has a 4a^2, that just
has a single a. So what I can do to this guy
right here is I can multiply the top and bottom
of that one by 4 times a. So we'll just make
a little note. This will be 4 times a, the
top will also be 4 times a. Alright, now I'm
going to write this just to make sure everything
is ordered nicely as -4ac and in the bottom
I'll have 4a^2. That looks pretty good. And
here is b^2 all over 4a^2. Ok, so that's all
looking good. In fact we are already starting
to see pieces of the quadratic formula show
up, there is b^2 there is -4ac. Its all looking
great. But anyway, the important part is that
we have that common denominator so we'll go
ahead an combine these into a single fraction
all over 4a^2. I like it. And as long as we
do have these into a single fraction, you
know the order of addition doesn't really
matter, so what I'm going to do, is I'm just
gong to flip flop these around. I'm going to
put my b^2 over here, I'm going to but the
4ac over there. So we'll say this is b^2 -4ac
all over 4a^2. Now I haven't touched the left
side, and of course we are just holding on
to that, we'll get to that in the next step,
its really just there, we'll get to it. But
you can see, yea, there is that quadratic
formula, we are starting to building it. Now
let's go ahead and work with the left side,
get rid of that square, and get this just
a little bit closer.
Alright so the next step, what shall we do
now? Well we are going to take the square
root of both sides. Now when you take the
square root of a variable that's already squared
in here, not only will it get rid of the square,
so we'll just end up with x + b/2a, but we
actually end up with a plus or minus possibility,
you know it could have been a positive variable,
it could have been negative, we don't know.
So we have both of those that we have to take
into account. Alright, looking better and
better. Now we have our square root in there,
and on this side where we have the square
root we can go ahead and take the square root
of the top, and the square root of the bottom.
Just to make it a little bit nicer.
Now the top is a fairly ugly complicated looking thing,
so I'm just going to leave it as the square
root of b^2 - 4ac, that looks familiar. The
bottom however, those are fairly nice numbers,
we can go ahead and take the square root of
that. The square root of 4 is just a 2, the
square root of a^2 is a. So that's not a problem.
Alright, and now we are almost there.
So in the next step let's go ahead and work
on getting this x all by itself. Let's move
that b/2a to the other side. Since its added
on the left we will subtract to get it to
the other side. Alright, now we are almost
home free. We've isolated the x, so that's
pretty good. These are essentially two fractions,
and they already have the same denominator.
That's great. That means we can actually take
these two factions and put them together so
that they are all just one single fraction.
In fact what fraction does that make? Let's
find out. Negative b plus or minus the square
root of b squared minus 4ac, all over the
common denominator of 2a.
And sure enough there is our quadratic formula.
And you can see that it really is just a process
of running through completing the square,
but doing it very carefully, every single
step. Making sure you simplify, making sure
you get that common denominator, and there you go.
You have the quadratic formula.
Alright, hopefully you enjoyed this video,
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