[Prepare to be astonished.] I need to show
you another way to solve quadratic
equations and it's going to be called
the quadratic formula.
Now we already have one way. We use the
zero property. Remember the zero property? We get a zero on one side of the
equation and then we factor to create
multiplication and when two things
multiply to zero, it can only be true when
one of them is zero. So we figure out what
x needs to be in each of the factors to
make each factor zero.
Let me show you how we
did it. For instance, in this equation we
got a 0 on one side by subtracting 15
from both sides.
Ok, now we have to create multiplication
so we factored. You have to trust me on the
factoring. -3 times 4 is -12
and 5 times x is 5. So a -12 and 5
give me the -7.
Okay, now I'm going to check to see what x
needs to be to make each of these
separate factors be 0. The x - 3, if I
solve it, x has to be 3 to make that
be 0. Of course I could have done that in my
head, couldn't I have. Some of them are a
little bit more difficult though. The 4x + 5, if I solve that to be zero,
I'll subtract 5 from both sides and
then divide by 4 and that answer is
-5/4. Note that these all have two
answers, don't they?
This one has two answers: x is 3 and x is
-5/4 and that was how we
use the zero property to solve quadratic
equations.
Let me show you this new way.
The quadratic formula looks like this.
Wait a minute. Don't go crazy. Any
quadratic equation is of the form
ax squared + bx + c where a is the
number in front of x squared. B is the
number in front of X and C is the number,
the constant if you would, sitting by
itself.
That's true. Don't get nervous. And if we
find out what negative of b plus or minus
the square root of b squared minus 4ac
all over 2a is, that's two answers
because of the plus or minus, we will
have those two answers.
Let me show you how we
do it.
We're are going to do the same problem
using the quadratic formula.
Remember, these were the two answers we
got before using the zero property. We
better get those same two answers.  First
as always, I'm going to get a 0 on one
side. You don't have an a, b and c until you
have a zero. So I'll subtract 15 from
both sides and there we go. Now a, the
number in front of x squared, is 4. B, the
number in front of x, is -7 and
C, is the number sitting by itself, is
-15.
Okay,
wait a minute. Now, we're going to take
those values and put them in to the
quadratic formula which is this.
You don't have to memorize it. You just
have to know how to use it.
negative of negative b, or negative of
-7 if you would, and then I
put a -7 in for b and a 4 in
for a and a -15 in for c and let's
see what we get. I'm going to get 7 plus
or minus the square root of 49.
Remember the 49 came from the -7
squared plus 240. Now where did that come
from? -4 times 4 times -15 is 240 all over 8. That's 2 times
4, right?
Ok, now 49 plus 240 is 289. Now, wait a minute.
The square root of 289 is 17. Now there's
two square roots. there's positive 17 and
negative 17. So we can do 7 + 17 over 8
and 7 - 17 over 8. 7 + 17 over 8 is 24/8.
7-17 over 8 is -10/8. Wait a
minute. Wait a minute. 24/8 is 3 and -10/8
is -5/4, so we really did
get the same answers, didn't we?
How cool is that?
I know what you're saying.
Why would we go to all that trouble to
get the same two answers that we could
have gotten just using the zero property?
Well, let's try this one with your
beloved zero property. We know we need to
get a zero on one side, so we'll add 3 to
both sides and now what is it time to do?
We have to factor, right? Okay, well go
ahead and factor it. What two
numbers multiply  to 3 and add up to 5? Come on. I'm waiting.
I'ts not factorable. It's not
factorable, and in fact,
most equations - we've been
spoiling you - most equations are not
factorable. You can keep trying to factor
until the cows come in.
It's not going to be able to be factored.
So what do you do? That's why we need
the quadratic formula. a will be 1, in
this case, because there's nothing there.
b will be 5 and c will be the
number sitting by itself, 3.
Let's put these numbers in. They're not going to
be pretty, but it's going to work.
Let's see. -5 plus or minus etc.,
is going to give me -5 plus or
minus 25 - 12. The square root of
 25 - 12 over 2 which is
-5 plus or minus the square root of 13 over 2. You understand there's two answers.
-5 plus the square root of 13 over
2 and -5 minus the square root
of 13 over 2. Got it? We like to write
it shorter, so instead of writing both of
those, we're going to write -5 plus or minus the square root of 13 over 2.
It just shortens our work. So get used to
seeing that.
Oh, that's the cows coming in. Okay, let's do another one. Sometimes it's going to take even a little
bit more work. Well, in this case, a is 2,
b is -2 and c is -1. This
seems pretty straightforward. We put it
into our quadratic formula and we get
the following.
Well, 2 plus or minus
the square root of 12 over 4 seems like
it definitely is two answers, but it's
not in simplest form and you're required
to put radicals, remember, in simplest
form. Let's do it.
The square root of 12 can be broken up
into its perfect squares and whatever is
left over. So
break it up into its perfect square, 4, and the garbage leftover,
3. Now the square root of 4 is 2
Ok, so I've simplified the radical. Now I
need to simplify the fraction. I note
that the top can be factored, okay. Don't
even think of canceling things that are
added though. That's not the way to 
simplify fractions. We're going to factor
the top. There's a 2 that comes out of both
of those things.
Okay, I divided 2 by 2 and I divided 2
radical 3 by 2. Now I can cancel the 2
with the 4 and I get my final answer.
And that, actually, is my final answers -
two answers. Get the idea? I
want you to understand these are two
answers: 1 plus the square root of 3 over
2 and 1 minus the square root of 3 over 2.
 
Now, the square root, we can get an
approximate answer if we needed on the
on the calculator, to make something
usable in real life. the square root of
3 is about 1.732. So 1 plus 1.732
over 2 and that's approximately
one of the answers. The other answer, 1
minus the square root 3 over 2, would be
1 minus 1.732 over 2 which would be -.732 over 2 which would come out to be
approximately -.366. I don't
want you to think these are some magical
numbers that are not useful in real life.
Okay, so we can use the quadratic formula
all the time, but generally we just use
it when we can't factor.
Ok, go try it. Do that homework.
