[And now for something completely different]
I want to show you another method for
solving basically non-factorable
quadratic equations and it's called
completing the square.
It's kind of wild.
If you can't factor, we
can't use the zero property, right. Well
ask yourself, as crazy as it may seem,
what number do you wish this was? Believe
it or not, you can do that. What number do
you wish it was so that you could factor
it?
Well I propose to you, that you wish that
that was the number 9.
[I don't understand] Why would I wish it
was 9? If it was 9,
just for a second, if it was 9, what
could we do? [Factor Boy] We could factor it. Not only
could we factor it, but the factors
would be the same. They would be x + 3 and
X + 3. They would be, if you would, a
perfect square. Now why am I always
trying to get a perfect square? Because
if I have a square, I can use an old
trick - the rooter. Ok and once I've used
the rooter I know how to do it. Of course, I
have to do it on both sides and I would
go on from there. But, we have to get back
to reality. You can't just erase that
-1 and make it a 9.
Actually that's basically what we do
when we create, or complete, a square.
The first thing we do is get rid of the
wrong number.
So let's get that out of there, legally,
by adding something to both sides. I'll
add 1 to both sides and now
he'll be gone. Okay, there won't be a zero
anymore, but remember when we use the
rooter we don't need a zero. We're not
using the zero property, are we? Now I
want to make a 9occur on that left
side and I can do it legally by adding a
9 to both sides.
Okay and I did a 9 to both sides and now I
have completed a square on the left side.
if I factor it,
Recall that it is x + 3 times x + 3.
I skipped a step there, but it is really
x + 3 squared.
Why did I want to do that? Because now
it's a perfect square and I can use the
rooter and if I root both sides,
on the left side, I'll get rid of
that 2 with the root and I'll have x + 3.
On the right side, recall that not
only do I get the square root of 10,
but plus or minus the square
root of 10. And now to solve for x
all I have to do is get rid of that
3 with his opposite of -3 and
I'll have
what x is. I solve this, if you
would, by completing the square.
And there are, in
fact two answers, aren't there, -3
plus radical 10 and -3 minus
radical 10. Now the first time I saw this
done,
the first thing I asked ,
how
did I know that 9 was the magic number?
How that I know that? Well there's a
trick. Always take the middle number, in
this case, 6. Divide it by 2. 6 divided by 2 is what? 3 and then
take that result
and square it and that's how you get the
number that you wish it was. In this
example, we'll take the middle number,
-8, divide it by 2.
What do we get? -8 divided by 2 is
4, -4 actually, but that doesn't really
matter because when we square it,
we're always going to get a positive
number, aren't we. So our wish number, in
this case, would have been 16. Well, let's
solve this by completing the square and
get that 16. I like to do it the same
way every time.
Let's get rid of the bad number. How are we
gonna do that legally? Subtract it from
both sides.
Okay. Now
I'm going to add the magic number.
Remember how I got that. I divided the
8 by 2, got 4, and squared it and
I'm going to add 16 because that's the number
I want.
Ok now, I'm promising you if you do that
that the left side will be a square.
Of course, you'll have to factor.
Ok, if you factor it you will get x - 4
times x - 4, known downtown on
Friday nights, as x - 4 squared. We've
completed a square.
Why did we do that? Now i can use the
rooter and don't forget, when you root a
number, you get two answers.
So the answer, in this case, of course we
have to get rid of the 4,
solving for x,
is x equals 4 plus or minus the square
root of 13. Not so bad. Now I've been
spoiling you a little bit. I want to give
you a little bit more difficult of a
problem.t
We can do it. Hang in
there. What's different about this
problem? Well, one thing,
you might think that we wish that was a
9, ok, by dividing 6 by 2 and squaring
it. But that's not going to work
unless this is a 1. Okay, unless
there's nothing there, in other words. So
when we're completing the square
actually, the first thing we're going to
have to do, the first thing, if there's
something in front of the x squared,
we're gonna have to get rid of that
first or our little dividing by 2 squaring trick
won't work.
Well, what we're going to
do, how we will get rid of him legally, is
divide him by 2. Of course, if I divide
him by 2, I have to do it to everybody and
that's what I'll do. That leaves fractions,
but quit crying, okay. Hang in there with
me. We can handle fractions.
Well what do we wish this was? Now our
trick will work.
Divide the middle number by 2 and square it.
Dividing 3 by 2, or -3 by 2, and
squaring it.
Let's see. I'm going to get rid of the
old number and now divide that by 2,
square that. -3 over 2 squared is
9/4 . So,
I'm going to add 9/4 to both
sides with the promise that I will have
completed a square. I know this is a
tough one, but hang with me. I'm going to add
9/4 to both sides and just in case
you're nervous, you forgot how to add
fractions. On the right side,
-5/2 is really -10/4. We
had to get a common denominator, and
-10 and 9 makes -1/4.
And on the left side, we just placed our
9/4. Well the promise is that if
you do that, if you divide the middle
number by 2 and square it and get that 9/4,
that if you factor it, apparently
you will be able to factor it, and
it'll be as perfect square.
Well, 9/4 did complete the square.
If we factor this, it's x - 3/2
times x - 3/2 and note
that if you have trouble factoring, that
it's always x minus or plus the middle
number divided by 2. That works every time.
The middle number divided by 2, in this case
3 divided by 2, okay. Now we've completed a square, as
ugly as it may seem,
okay. And what do we do? We'll take the
square root of both sides to get rid of
that square,
ok which it will do. And on the right
side, how did I know i was involved
there? Because it's a square root of a
negative number. Now, I took the square
root of 1 and got 1. I took the square
root of 4 and got 2 and that
negative, which is really -1, is
the square root of -1 is i, okay.
Now, how do I solve for x?
Oh, don't forget. We have a plus or minus
any time you take a square root. See
even I forgot.
Okay, so I'm going to solve for x by adding
3/2 to both sides
and in this problem, and I don't
think they get any harder than this,
the answers are 3/2 plus or
minus 1/2i. We solved that
by completing the square. I don't know
any other way that you could
have done it.
Okay, remember the trick. Take the middle
number, divide it by 2 and square it, and
that's what you want the constant to be.
Don't worry. You'll never have to do
something that bad again.
But, you do have to go do your
homework. Go ahead and try and complete
the square.