>> We're going to talk about
the discriminant in this video.
So if you look at the quadratic
formula x equals negative b plus
or minus the square root of b
squared minus 4c all over 2a,
the discriminant is this part
underneath the square root
symbol; the b square minus 4ac.
So it's defined to be
exactly that which,
if you plug in the values of
a, b, and c you'll get a number
so the discriminant is the value
of the b squared minus 4ac.
Now if you plug in values
there's only 3 things
that can happen.
Either it's greater than 0
so it's a positive number,
or it's equal to 0
or it's less than 0
which means it's
a negative number.
So if you think about the
quadratic formula, or just look
in the numerator, it's
going to be negative b plus
or minus the square root of
some positive number okay?
All over 2a, so if you think
about what's going to go
on whatever happens to be,
whatever b is, you're going
to add some positive number
and then it will be different
than when you subtract
some positive number.
Actually it's going
to be over 2a right?
So when the discriminant is
greater than 0 you're going
to get 2 different solutions and
you're taking the square root
of a positive number so it's
going to be a real number.
So when b squared minus 4ac
is greater than 0 you're going
to have 2 real numbers; 2
real numbers for solutions.
Okay? Let's bring these down;
don't want to get confused.
Okay now what about if b
squared minus 4 ac equals 0?
Well, what would that look like?
You'd have negative b plus
or minus the square root of 0
over 2a so basically you're
going to get negative b plusser
or the negative b minus;
that's the same thing.
You're going to get
negative b over 2a.
This happens when whatever
you have is the perfect square
basically and you
could have factored it.
So you're only going to
get one real solution;
because you're not going
to get any negative numbers
or square roots so you're
going to get one real solution.
In fact instead of
saying 2 real numbers,
I'm going to say
2 real solutions.
All right now what happens
if what's underneath the square
root is a negative number?
So you've got some
negative number over 2a.
Well you're going
to get negative--
that's going to give
you i right?
Because you take the square
root of some negative number
so you're going to do plus
something with an i in it
and then minus something
with an i in it; you're going
to get two different
solutions; in fact they're going
to be complex conjugates.
So you're going to get
two complex conjugates
for solutions; two complex
solutions and I'm going
to say a little bit
more about this.
They're actually
conjugates, complex conjugates
and they're not real
that means right?
Just keep in mind they're not
real; two complex solutions.
So the discriminant tells you
ahead of time if your solutions,
if you just check out
what the discriminant is,
you know if your solutions are
going to be two real solutions,
one real solutions or
two complex solutions.
And this is going to help
us when we do graphing.
It's going to be
important to be able to tell
when you're graphing something
whether it goes to the x axis.
More about that later, but for
now the discriminant will tell
us something about
our solutions.
So let's do a couple
of problems.
All right so we're going
to use the discriminate
to determine the
number and types
of solutions of each equation.
So here's the first one--
2x squared minus
3x plus 4 equals 0.
Well you could use the quadratic
formula, the whole thing,
and then at the end analyze it
and say hey two real solutions,
one real solutions, or two
complex not real solutions.
Or the easy way simply
do the discriminate
because it's not asking
us for the solutions;
it's only asking us for
the types of solutions.
So let's just do that.
B squared minus 4ac
in this case.
Well b squared negative 3 times
negative 3 is 9 minus 4 times
ac; 2 times 4 is 8.
And you don't even have to
figure out what number that is
but all you have to do
is decide is that going
to be a negative number,
a positive number or a 0?
So think about it 9 minus 32.
Is that going to be
positive or negative?
It's going to be negative.
So what happens?
The discriminant is
negative so the part
under the square root will
be negative; so you are going
to get two complex which
means they're not real right?
Solutions.
And they're actually
two complex conjugates.
Does that make sense to you?
The part under the square root
in the quadratic formula
would be negative so just
to show you once again.
It's because you have x equals
negative b if you were going
to do the whole thing.
You'd have 3 plus or
minus the square root
of some negative number right?
Over 2a; all right you know
you're going to have some i
in there some place and that's
why it ends up being complex.
Okay? All right let's do
a different problem now.
All right so let's find
out what kind of solutions
without actually solving this
and writing the solutions,
what are the-- are the
solutions going to be real,
two or one of them, or are they
going to be a complex conjugate?
All right well we need to
put this in standard form
so we can identify
a, b, and c. A is 4,
b is negative 12, c is 9.
So I want to know what
is b squared minus 4ac,
that's the discriminant;
so b squared negative 12 times
negative 12 is 144 minus 4
times ac.
4 times 9 is 36.
Okay so I've got 4 times
36 is 144 oh I get 0.
Hm what did that mean?
That means if I was
going to plug it
in into the quadratic formula,
I'd have plus the square to 0
and minus the square to 0.
So I'm going to get
one real solution.
Now why is that?
Look at that original problem
4x squared minus 12x plus 9.
Somebody said to solve this
and you just factored it.
It's a perfect square;
even if you wrote it
as 2x minus 3 times 2x minus 3.
So if you solved
it, you would end
up getting just one solution.
2x is 3 or x is 3 halves.
So if you were going to solve
it, you could see why it ends
up only being 1 real solution.
But the idea here is you're not
having to go through all that.
All it's asking you is
about the types of reaps
and the types of solutions.
Reap is another word
for solutions in math.
So it has one real solution.
That will happen anytime
it could have been factored
because it was a perfect square.
Let's try to get
one more in here.
Let's find out about
these roots.
A is 8; B is negative
2; C is negative 7
and you all know you could
always put the video on pause
and try the problem
before I do it.
So let's do b squared minus 4ac.
B squared is negative, 2 times
negative 2 that's 4 minus 4
times ac.
So I've got 8 times negative
7, that's negative 56.
All right now let's
just look at that?
Is this going to be a
positive or negative number?
It's going to be 4
plus something right?
It's greater than 0 I
don't care what it is.
But that means if you did
the quadratic formula,
if you were actually having
to solve and get the solution,
you would end up with
plus the square root
of some positive number
and minus the squared
of some positive number.
You've got negative
b plus that number,
and negative b whatever
it is minus that number.
So this one is going
to be 2 real solutions.
Let me ask you one more thing?
What if the discriminant
was a positive square?
What if b squared minus 4ac?
What if b squared minus
4ac was a perfect square?
Like 36. You'd have
plus the square to 36
and minus the square to 36
so you know what it ends
up being a perfect square.
They're going to be
too rational solutions.
You won't even have
any square roots.
So that tell you even a
little bit more information;
just in case you're interested.
