Welcome to the video on completing the square.
What's completing the square?
Well, it's a way to solve a quadratic equation.
And so before I--actually, let me just write down a quadratic equation, and
then I will show you how to complete the square.
And then we'll do another example, and then maybe we'll talk
a little bit about why it's called completing the square.
So let's say I have this equation: x squared plus 16x
minus 57 is equal to 0.
So what are the tools in our toolkit right now that we
could use to solve this?
Well, we could try to factor it out.
We could say, what two numbers add up to 16, and then when you
multiply them they're minus 57?
And you'd have to think about it a little bit.
And you might get whole numbers, but you're not even
sure if there are two whole numbers that work
out like that.
This problem there are.
But, you know, sometimes the solution is a decimal number
and you don't know it.
So the only time you can really factor is if you're sure that
you could factor this into kind of integer expressions.
You know, x plus some integer or x minus some integer
times, you know, x plus some other integer.
Or likewise.
The other option is to do the quadratic equation.
And what we're going to see is actually the quadratic equation
is just essentially a shortcut to completing the square.
The quadratic equation is actually proven using
completing the square.
So what is completing the square?
So what do we do?
Well, before we move into this video let's see what happens
if I square an expression.
Let me do it in this down here.
What is x plus a, squared?
Well that equals x squared plus 2ax plus a squared.
Right?
So if you ever see something in this form, you know that it's
x plus something squared.
So wouldn't it be neat if we could manipulate this equation
so we can write it as x plus a squared equals something,
and then we could just take the square root?
And what we're going to do is, actually, do just that.
And that is completing the square.
So let me show you an example.
I think an example will make it a little clearer.
Let me box this away.
This is what you need to remember.
This is the whole rationale behind competing the squares--
to get an equation into this form, onto one side of the
equation, and just have a number on the other side, so
you could take the square root of both sides.
So let's see.
First of all, let's just check to make sure this isn't
a perfect square.
If this were, this coefficient would be equivalent to the 2a.
Right?
So a would be 8, and then this would be 64.
This is clearly not 64, so this right here is not
a squared expression.
So what can we do?
Well let me get rid of the 57 by adding 57 to both
sides of this equation.
So I would get x squared plus 16x is equal to 57.
All I did is I added 57 to both sides of this equation.
Now, what could I add here so that this, the left-hand side
of this equation, becomes a square of some expression
like x plus a?
If you just follow this pattern down here, we have x squared
plus 2ax-- so you could view this right here as 2ax.
Right?
That's 2ax.
And then we need to add an a squared to it.
Right?
Plus a squared.
And then we would have the form here.
But we know from basic algebra that anything you do to one
side of an equation you have to do to another.
So we added an a squared here, so let's add an a
squared here as well.
And now you could essentially rewrite this as a square
of some expression.
But before that we have to figure out what a was?
Well how do we do that?
Well, what is a?
If this expression right here is 2ax, what is a?
Well 2a is going to equal 16, so a is equal to 8.
And you could usually do that just by inspection;
do it in your head.
But if you wanted to see it done algebraically you could
actually write 2ax is equal to 16x.
And then divide both sides by 2x, and you get a is
equal to 16x over 2x.
And assuming that x doesn't equal 0 this evaluates to 8.
So a is 8.
So if a is 8 we could rewrite that expression-- I'll switch
colors arbitrarily-- as x squared plus 16x
plus a squared.
Well, it's 64, because a is 8.
Is equal to 57 plus 64.
Right?
I went through a fairly tedious explanation here, but all we've
really done to get from there to there is we added 57 to both
sides of this equation to kind of get it on the right-hand
side, and then we added 64 to both sides of this equation.
And why did I add 64 to both sides of this equation?
So that the left-hand side expression takes this form.
Now that the left-hand side expression takes this form
I can rewrite it as what?
x plus a, squared.
I can rewrite it in this form.
And we know that a is 8, so it becomes x plus 8, squared,
is equal to-- and what's 57 plus 64?
It's 121.
Now we have what looks like a fairly straightforward-- it's
still a quadratic equation, actually, because if you
were to expand this side you'd get a quadratic.
But we can solve this without using the quadratic equation
or without having to factor.
We can just take the square root of both sides of this.
And if we take the square root of both sides what do we get?
We get-- once again, arbitrarily switching colors--
that x plus 8 is equal to, and remember this, the plus or
minus square root of 121.
And what's the square root of 121?
Well it's 11, right?
So then we come here.
Let me box this away.
This was just an aside.
So we get x plus 8 is equal to plus or minus 11.
And so x is equal to-- subtract 8 from both sides-- minus
8 plus or minus 11.
And so x could equal-- so minus 8 plus 11 is 3.
Right?
Let me make sure I did that right.
x is equal to minus 8 plus or minus 11.
Yes.
That's right.
So x could be equal to 3.
And then if I took minus 8 minus 11, x could
also equal minus 19.
All right.
And let's see if that makes sense.
So in theory this should be able to be factored as x
minus 3 times x plus 19 is equal to 0.
Right?
Because these are the two solutions of this equation.
And that works out, right?
Minus 3 times 19 is minus 57.
And minus 3 plus 19 is plus 16x.
We could have just immediately factored it this way, but if
that wasn't obvious to us-- because, you know, at least
19 is kind of a strange number-- we could do it
completing the square.
And so why is it called completing the square?
Because you get it in this form and then you have to add this
64 here to kind of complete the square-- to turn this
left-hand expression into a squared expression.
Let's do one more.
And I'll do less explanation and more just chugging through
the problem, and that actually might make it seem simpler.
But this is going to be a hairier problem.
So let's say I have 6x squared minus 7x minus 3 is equal to 0.
You could try to factor it, but personally I don't
enjoy factoring things when I have a coefficient.
And you can say, oh well why don't we divide both sides
of this equation by 6?
But then you'd get a fraction here and a fraction here.
And that's even worse to factor just by inspection.
You could do the quadratic equation.
And maybe I'll show you in a future video, the quadratic
equation-- and I think I've already done one where I proved
the quadratic equation.
But the quadratic equation is essentially
completing the square.
It's kind of a shortcut.
It's just kind of remembering the formula.
But let's complete the square here, because that's what the
point of this video was.
So let's add the 3 to both sides of that equation.
We could do-- well, let's add the 3 first.
So you get 6 x squared minus 7x is equal to 3.
I added 3 to both sides.
And some teachers will leave the minus 3 here, and then try
to figure out what to add to it and all of that.
But I like to get it out of the way so that I can figure out
very clearly what number I should put here.
But I also don't like the 6 here.
It just complicates things.
I like to have it x plus a squared, not some square root
coefficient on the x term.
So let's divide both sides of this equation by 6, and you get
x squared minus 7/6 x is equal to-- 3 divided by 6
is equal to 1/2.
And we could have made that our first step.
We could have divided by 6 right at that first step.
Anyway, now let's try to complete the square.
So we have x squared-- I'm just going to open up some space--
minus 7/6 x plus something is going to be equal to 1/2.
And so we have to add something here so that this left-hand
expression becomes a squared expression.
So how do we do that?
Well essentially we look at this coefficient, and keep
in mind this is not just 7/6 it's minus 7/6.
You take 1/2 of it, and then you square it.
Right?
Let me do it.
x plus a, squared, is equal to x squared plus
2ax plus a squared.
Right?
This is what you have to remember all the time.
That's all completing the square is based off of.
So what did I say just now?
Well, this term is going to be 1/2 of this
coefficient squared.
And how do we know that?
Because a is going to be 1/2 of this coefficient if you just
do a little bit of pattern matching.
So what's 1/2 of this coefficient?
1/2 of minus 7/6 is minus 7/12.
So if you want you could write a equals minus
7/12 for our example.
And I just multiplied this by 1/2.
Right?
So what do I add to both sides?
I add a squared.
So what's 7/12 squared?
Well that's going to be 49/144.
If I did it to the left-hand side I have to do it to
the right-hand side.
Plus 49/144.
And now how can I simplify this left-hand side?
What's our next step?
Well we now know it is a perfect square.
In fact, we know what a is. a is minus 7/12.
And so we know that this left-hand side of this equation
is x minus a-- or x plus a, but a is a negative number.
So x plus a, and a is negative, squared.
And if you want you can multiply this out and confirm
that it truly equals this.
And that is going to be equal to-- let's get a common
denominator, 144.
So 72 plus 49 equals 121.
121/144.
So we have x minus 7/12, all of that squared
is equal to 121/144.
So what do we do now?
Well now we just take the square root of both
sides of this equation.
And I'm trying to free up some space.
Switch to green.
Let me partition this off.
And we get x minus 7/12 is equal to the plus or minus
square root of that.
So plus or minus 11/12.
Right?
Square root of 121 is 11.
Square root of 144 is 12.
So then we could add 7/12 to both sides of this equation,
and we get x is equal to 7/12 plus or minus 11/12.
Well that equals 7 plus or minus 11/12.
So what are the two options?
7 plus 11 is 18, over 12.
So x could equal 18/12, is 3/2.
Or, what's 7 minus 11?
That's minus 4/12.
So it's minus 1/3.
There you have it.
That is completing the square.
Hopefully you found that reasonably insightful.
And if you want to prove the quadratic equation, all you
have to do is instead of having numbers here, write a x squared
plus bx plus c equals 0.
And then complete the square using the a, b, and c's
instead of numbers.
And you will end up with the quadratic equation
by this point.
And I think I did that in a video.
Let me know if I didn't and I'll do it for you.
Anyway, I'll see you in the next video.