Quadratic formula (proof) | Quadratic equations | Algebra I | Khan Academy

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.