Welcome to part two of
the presentation on

quadratic equations.

Well, I think I thoroughly
confused you the last time

around, so let me see if I
can fix that a bit by doing

several more examples.

So let's just start with
a review of what the

quadratic equation is.

The quadratic equation says, if
I'm trying to solve the

equation Ax squared plus Bx
plus C equals 0, then the

solution or the solutions
because there's usually two

times that it intersects the
x-axis, or two solutions for

this equation is x equals minus
B plus or minus the square root

of B squared minus
4 times A times C.

And all of that over 2A.

So let's do a problem and
hopefully this should make

a little more sense.

That's a 2 on the bottom.

So let's say I had the equation
minus 9x squared minus

9x plus 6 equals 0.

So in this example what's A?

Well, A is the coefficient
on the x squared term.

The x squared term is here,
the coefficient is minus 9.

So let's write that.

A equals minus 9.

What does B equal?

B is the coefficient on the x
term, so that's this term here.

So B is also equal to minus 9.

And C is the constant term,
which in this example is 6.

So C is equal to 6.

Now we just substitute these
values into the actual

quadratic equation.

So negative B, so it's
negative times negative 9.

That's B.

Plus or minus the square root
of B squared, so that's 81.

Right?

Negative 9 squared.

Minus 4 times negative 9.

That's A.

Times C, which is 6.

And all of that over 2
times negative 9, which

is minus 18, right?

2 times negative 9-- 2A.

Let's try to simplify
this up here.

Well, negative negative
9, that's positive 9.

Plus or minus the
square root of 81.

Let's see.

This is negative 4
times negative 9.

Negative 4 times negative
9 is positive 36.

And then positive 36
times 6 is-- let's see.

30 times 6 is 180.

And then 180 plus
another 36 is 216.

Plus 216, is that right?

180 plus 36 is 216.

All of that over 2A.

2A we already said is minus 19.

So we simplify that more.

That's 9 plus or minus the
square root 81 plus 216.

That's 80 plus 217.

That's 297.

And all of that over minus 18.

Now, this is actually-- the
hardest part with the quadratic

equation is oftentimes just
simplifying this expression.

We have to figure out if we
can simplify this radical.

Well, let's see.

One way to figure out if a
number is divisible by 9 is to

actually add up the digits
and see if the digits

are divisible by 9.

In this case, it is.

2 plus 9 plus 7 is equal to 18.

So let's see how many
times 9 goes into that.

I'll do it on the side here; I
don't want to be too messy.

9 goes into 2 97.

3 times 27.

27-- it goes 33 times, right?

So this is the same thing as 9
plus or minus the square root

of 9 times 33 over minus 18.

And 9 is a perfect square.

That's why I actually wanted to
see if 9 would work because

that's the only way I could get
it out of the radical, if

it's a perfect square.

As you learned in that exponent
rules number one module.

So this is equal to 9 plus
or minus 3 times the square

root of 33, and all of
that over minus 18.

We're almost done.

We can actually simplify it
because 9, 3, and minus 18

are all divisible by 3.

Let's divide everything by 3.

3 plus or minus the square
root of 33 over minus 6.

And we are done.

So as you see, the hardest
thing with the quadratic

equation is often just
simplifying the expression.

But what we've said, I know you
might have lost track-- we did

all this math-- is we said,
this equation: minus 9x

squared minus 9x plus 6.

Now we found two x values that
would satisfy this equation

and make it equal to 0.

One x value is x equals
3 plus the square root

of 33 over minus 6.

And the second value is
3 minus the square root

of 33 over minus 6.

And you might want to
think about why we have

that plus or minus.

We have that plus or minus
because a square root could

actually be a positive
or a negative number.

Let's do another problem.

Hopefully this one will
be a little bit simpler.

Let's say I wanted to
solve minus 8x squared

plus 5x plus 9.

Now I'm going to assume that
you've memorized the quadratic

equation because that's
something you should do.

Or you should write it
down on a piece of paper.

But the quadratic equation is
negative B-- So b is 5, right?

We're trying to solve that
equal to 0, so negative B.

So negative 5, plus or minus
the square root of B squared-

that's 5 squared, 25.

Minus 4 times A,
which is minus 8.

Times C, which is 9.

And all of that over 2 times A.

Well, A is minus 8, so all
of that is over minus 16.

So let's simplify this
expression up here.

Well, that's equal to
minus 5 plus or minus

the square root of 25.

Let's see.

4 times 8 is 32 and the
negatives cancel out, so

that's positive 32 times 9.

Positive 32 times 9, let's see.

30 times 9 is 270.

It's 288.

I think.

Right?

288.

We have all of that
over minus 16.

Now simplify it more.

Minus 5 plus or minus the
square root-- 25 plus

288 is 313 I believe.

And all of that over minus 16.

And I think, I'm not 100% sure,
although I'm pretty sure.

I haven't checked it.

That 313 can't be factored
into a product of a perfect

square and another number.

In fact, it actually
might be a prime number.

That's something that you
might want to check out.

So if that is the case and
we've got it in completely

simplified form, and we say
there are two solutions, two

x values that will make
this equation true.

One of them is x is equal
to minus 5 plus the square

root of 313 over minus 16.

And the other one is x is equal
to minus 5 minus the square

root of 313 over minus 16.

Hopefully those two examples
will give you a good

sense of how to use the
quadratic equation.

I might add some more modules.

And then, once you master this,
I'll actually teach you how to

solve quadratic equations when
you actually get a negative

number under the radical.

Very interesting.

Anyway, I hope you can do the
module now and maybe I'll add a

few more presentations because
this isn't the easiest module.

But I hope you have fun.

Bye.