In the completing the square video I kept saying that all
the quadratic equation is completing the square
as kind of a short cut of completing square.
And I was under the impression that I had done this
proof already but now I realize that I haven't.
So let me prove the quadratic equation to you, by
completing the square.
So let's say I have a quadratic equation.
I guess a quadratic equation is actually what you're trying to
solve, and what a lot of people call the quadratic equation is
actually the quadratic formula.
But anyway I don't want to get caught up in terminology.
But let's say that I have a quadratic equation that
says ax squared plus bx plus c is equal to 0.
And let's just complete the square here.
So how do we do that?
Well let's subtract c from both sides so we get ax squared plus
the bx is equal to minus c.
And just like I said in the completing the square video
I don't like having this a coefficient here.
I like just having one coefficient on my x squared
term so let me divide everything by a.
So I get x squared plus b/a x is equal to-- you have
to divide both sides by a --minus c/a.
Now we are ready to complete the square.
What was completing the square?
Well it's somehow adding something to this expression so
it has the form of something that is the square
of an expression.
What do i mean by that?
Well, I'll do a little aside here.
if I told you that x plus a squared, that equals
x squared plus two ax plus a squared, right?
So if I can add something here so that this left hand side
this expression looks like this, then I could
go the other way.
I can say this is going to be x plus something squared.
So what do I have to add on both sides?
If you watched the completing the square video this should be
hopefully intuitive for you.
What you do is you say well this b/a, this corresponds to
the 2a term, so a is going to be half of this, is going to
be half of this coefficient.
That would be the a.
And then what I need to add is a squared.
So I need to take half of this and then square it and
then add it to both sides.
Let me do that in a different color.
Do it in this magenta.
So I'm going to take half of this-- I'm just completing
square, that's all I'm doing, no magic here --so
plus half of this.
Well half of that is b/2a right?
You just multiply by 1/2.
And I have to square it.
Well if I did it to the left hand side of the equation, I
have to do it to the right hand side.
So plus b/2a squared.
And now I have this left hand side of the equation in the
form that it is the square of an expression that is
x plus something.
And what is it?
Well that's equal to-- let me switch colors again --what's
the left hand side of this equation equal to?
And you can just use this pattern and go to the left.
It's x plus what?
Well we said a, you can do one of two ways. a is 1/2 of this
coefficient or a is the square root of this coefficient or
since we didn't even square it we know that this
is a. b/2a is a.
So this is the same thing as x plus b over 2a everything
squared, and then that equals-- let's see if we can simplify
this or make this a little bit cleaner --that equals--
See, if I were to have a common denominator-- I'm just doing a
little bit of algebra here --see, when I square this it's
going to be 4a squared-- let me let me write this.
This is equal to b squared over 4a squared.
Right?
And so if I have to add these two fractions, let me make
this equal to 4a squared.
Right?
And if the denominator is 4a squared, what does
the minus c/a become?
I See if I multiply the denominator by 4a, I have to
multiply the numerator by 4a.
So this becomes minus 4ac, right?
And then b squared over 4a squared, well that's
just still b squared.
I'm just doing a little bit of algrebra.
Hopefully I'm not confusing you.
I just expanded this.
I just took the square of this, b squared over 4a squared.
And then I added this to this, I got a common denominator.
And minus c/a is the same thing as minus 4ac over 4a squared.
And now we can take the square root of both
sides of this equation.
And this should hopefully start to look a little
bit familiar to you now.
So let's see, so we get x.
So if we take the square root of both sides of this equation
we get x plus b/2a is equal to the square root of this-- let's
take the square root of the numerator and the demoninator.
So the numerator is-- I'm going to put the b squared first, I'm
just going to switch this order, it doesn't matter --the
square root of b squared minus 4ac, right?
That's just the numerator.
I just the square root of it, and we have to get the square
root of the denominator too.
What's the square roof of 4a squared?
Well it's just 2a, right?
2a.
And now what do we do?
Oh, it's very important!
When we're taking the square root, it's not just the
positive square root.
It's the positive or minus square root.
We saw that couple of times when we did the-- and you could
say it's a plus or minus here too, but if you look plus or
minus on the top and a plus or minus on the bottom, you can
just write it once on the top.
I'll let you think about why you only have to write it once.
If you had a negative an a plus, or negative and a plus
sometimes cancel out, or a negative and a negative,
that's the same thing as just having a plus on top.
Anyway, I think you get that.
And now we just have to subtract b/2a from both sides.
and we get, we get-- and this is the exciting part --we get x
is equal to minus be over to 2a plus or minus this thing, so
minus b squared minus 4ac, all of that over 2a.
And we already have a common denominator, so we can
just add the fractions.
So we got --and I'm going to do this in a vibrant bold-- I
don't know maybe not so much bold, well green color --so we
get x is equal to, numerator, negative b plus or minus square
root of b squared minus 4ac, all of that over 2a.
And that is the famous quadratic formula.
So, there we go we proved it.
And we proved it just from completing the square.
I hope you found that vaguely interesting.
See in the next Video.