OK.
In this problem, I need to solve
the quadratic equation.
And first of all, I just
want to point a couple
things out to you.
This is certainly an equation
because of this equals sign
right here.
And even though the prefix, or
the word quadratic, might make
you think of the number 4, it
actually has to do with this
largest exponent being a two.
So in an equation, if the
largest exponent is a 2 and
it's got an equal sign, you
have a quadratic equation.
So we can solve this thing in
several different ways.
The fastest way is to factor.
That's the fastest way.
So let's actually try that
first real quick.
What I'm going to do is I'm
going to take this 2 right
here, and I'm going to bring
it over from the right hand
side over to the
left hand side.
So if you change sides,
you change signs.
It's now a negative 2 on the
left hand side, which means we
have nothing remaining.
We have 0 remaining on
the right hand side.
OK.
So if I try to factor this real
quickly here-- if I try
to factor this left hand side,
I see a minus sign right here
at the very end, which means my
signs have to be different.
The only way to split up
5x squared is 5x and x.
The only way to split up
2 is to do 2 and 1.
[? Maybe ?] just put the 2
here and put the 1 there.
Now if I factored correctly--
all right?
If I factor correctly, I hope
you see I can get two answers
out of this pretty
quickly here.
Right?
I'm technically setting each
of these parentheses here
equal to 0.
This one here and this one,
technically set equal to 0
since it is an equation.
And so I get two answers of
positive 2, and I get, let's
see, another answer every
here of negative 1/5.
Hope you see how
I can do that.
But that's up to you to figure
out on your own.
But that's not really
what I'm after.
What I'm simply after is did
I even factor correctly?
Maybe I didn't even factor
correctly at all.
So we should probably
check that first.
In other words, if you were to
FOIL these two parentheses
that I have here
now, would I--
OK, this erasing is taking
way too long--
would I get back to the
trinomial that I have on the
left hand side of my equation?
Well let's go check and
see if that works.
I'm not going do the whole
full out FOIL.
I'm actually just going
to try the inners
and the outers here.
OK?
So the inners.
Let's see.
1 times x would give
me just x.
And 5x times negative 2 would
give me a negative 10x.
And if you add those up, you'd
see that we would get a
negative 9x.
Uh, man, that's not
what I wanted.
I wanted a positive 6x.
So I hope you see, and
I'm not going to
bother doing this anymore.
Look, factoring just isn't
going to work.
OK?
It's just not going
to work at all.
So let me get rid of all
of this stuff here.
And one method that always works
no matter what we try to
do is called the quadratic
formula.
Right?
Quadratic formula
always works.
And so you're going to have
to memorize that thing.
It is--
let's see if we can get it up
over here-- quadratic formula
is negative b plus or minus
the square root b
squared minus 4ac.
And as I'm writing this
here, it reminds me.
Do a search on YouTube for
quadratic formula song.
A bunch of songs have come up.
Maybe one of them or two of them
will catch your fancy.
And that's a good way to
learn a formula is to
commit it to song.
OK.
So we've got our quadratic
formula over here.
And I need to pick off the
players, a, b, and c, because
that's what we need inside
of this formula.
So let's go find out what
a, b, and c are.
I hope you see that a is simply
just the coefficient of
our squared term.
b, let's see, b over here is
just this plus 6, this
positive 6.
And c is negative 2.
If any of these things
are missing, they
would just be at 0.
By the way, a cannot be 0.
Think about that.
Wouldn't be quadratic.
All right.
We've got all the players now.
Let's plug them into
our formula.
And I read this part of the
formula here, the very first
thing, as give me the
opposite of b.
Since my b is a 6, I want the
opposite of that, which is
negative 6.
OK?
If my b was negative
6, then I would put
positive 6 to begin with.
All right.
So first thing is give me the
opposite of b, which is 6 plus
or minus the square root
of-- all right
now give me b squared.
That's b times itself, or 6
times 6 is 36 minus 4 times a
couple of numbers.
What are those numbers?
Well those numbers
are a and c.
a is 5 in this case, and c is a
negative 2, all over-- that
fraction bar goes all
the way across, OK?
Not just underneath
the radical stuff.
All the way across.
All over 2 times my a.
And my a is a 5.
OK, so I'm going to take this
bit by bit and simplify it
down as much as I can.
I hope you see and agree that,
look, this 2 times 5 on the
bottom, that's simply
just a 10, right?
We can do that right here.
We'll just put a 10 down here.
I still have that negative
6 sitting out front.
OK.
Let's go to work inside
this square root now.
Let's go work inside
the square root.
And keep in mind, if this
right here is larger--
all right, if all of this that
I put in a red box here is
larger than 36-- in other
words, 36 minus a bigger
number than itself--
that would give me a negative
number altogether underneath
the radical.
And you can't have a negative
number sitting
underneath the radical.
Well at least you can't in
lower level math classes
because that's not real.
In upper level math classes,
you can handle it with
imaginary numbers.
But we're not going to do deal
with imaginary numbers now.
OK?
So let me get rid of this
stuff here for now.
What is this number?
What's the result of what
I have in the red box?
Well, let's see.
4 times 5 is 20.
20 times negative 2
is negative 40.
But do you see there's a
negative sitting out in front
of this red box here?
So the negative here, the
minus sign here, and the
negative 40 that is sitting
inside the red
box, becomes a positive.
So I really have, I hope
you see, 36 plus 40.
Right?
So those two negatives cancel
each other out.
Hey, that's the number 76.
Cool.
So I have a 76 sitting
inside that radical.
Now you can look at a previous
video of mine, and you could
see how do we simplify 76.
And maybe over here off on the
side, you can break it down
into its prime factors.
You could take 76 and cut it
in half, right, and get 38.
That's prime.
Now let's see, what's
half of 38?
Half of 38 would be 19.
That's prime.
And 19 is a prime number.
You can't going further
with that.
So I hope you see, and you can
check this out in another
video of mine, that, look, I
have a pair of 2's underneath
that radical.
That's going to come out of
the radical as a single 2.
All right?
So here's what I've got.
I've got negative 6
plus or minus 2.
Again, that's coming from those
identical pair of 2's
right there, which
was from the 76.
And then 19 is still inside
the radical sign.
All over 10.
Now we would normally
stop right there,
except for one thing.
Let's play around with the
numbers that are outside of
the radical.
In other words, forget
the 19 because
that's inside the radical.
But look, see this
number here?
See that number right there?
See that number right there?
All of them, all three
of these, have
something in common.
They're all divisible by 2.
Right?
The'yre all even.
So I can simplify this answer
in just one more step.
Now, what I'm going to show
you is really, really
important in that--
here, let me get rid of
some stuff because I'm
running out of room.
But what I'm going to show you
is really, really important
because it only works
if-- all right?
It only works if all three of
these things-- all right, the
negative 6, the 2, and the
10-- all three of these
numbers that are not inside the
radical are divisible by
the same number.
That's the only way
this will work.
There's a mathematical reason
for behind it, but I'm not
going to go there just yet.
OK, so I'm going to cut
negative 6 in half.
That is a negative 3.
I'm going to cut 2 in half.
That is a 1.
My 19 is still underneath
the radical.
And I'm going to
cut 10 in half.
That is a 5.
So there are my two answers.
Negative 3 plus 1 radical
19, and negative 3
minus 1 radical 19.
Now mathematicians are kind of
lazy so we don't really even
need to put the 1
there sometimes.
We can just write the two
answers just like that.
Hope that helps.
