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## Discriminant and number of solutions

Current time:0:00Total duration:10:14

# Discriminant of quadratic equations

## Video transcript

I think we've had some pretty
good exposure to the quadratic formula, but just in case you
haven't memorized it yet, let me write it down again. So let's say we have a quadratic
equation of the form, ax squared plus bx,
plus c is equal to 0. The quadratic formula, which we
proved in the last video, says that the solutions to this
equation are x is equal to negative b plus or minus the
square root of b squared, minus 4ac, all of
that over 2a. Now, in this video, rather than
just giving a bunch of examples of substituting in the
a's, the b's, and the c's, I want to talk a little bit
about this part of the quadratic formula, this
part right there. The b squared minus 4ac. And we've seen it in a couple of
the problems we've done as examples, that this kind of
determines what our solution is going to look like. If, for example, b squared minus
4ac is greater than 0, we're going to have two
solutions, right? The square root of some
positive number that's non-zero, there's going to be
a positive and negative version of it-- we're always
going to have a b over 2a or negative b over 2a-- so you're
going to have negative b plus that positive square root, and
a negative b minus that positive square root,
all over 2a. So if the discriminant is
greater 0, then that tells us that we have two solutions. Now I just used a word, and
that word is discriminant. And all that is referring
to is this part of the quadratic formula. That right there-- let me do it
in a different color-- this right here is the discriminant
of the quadratic equation right here. And you just have to remember,
it's the part that's under the radical sign of the
quadratic formula. And that's why it matters,
because if this is greater than 0, you're having a positive
square root, and you'll have the positive and
negative version of it, you'll have two solutions. Now, what happens if b squared
minus 4ac is equal to 0? If this is equal to 0-- if you
take b squared minus 4, times a, times c, and that's equal to
0-- that tells us that this part of the quadratic formula
is going to be 0, and the square root of 0 is just 0. And then, actually, your only
solution is going to be x is going to be equal to
negative b over 2a. Or another way to think
about it is you only have one solution. So if the discriminant
is equal to 0, you only have one solution. And that solution is actually
going to be the vertex, or the x-coordinate of the vertex,
because you're going to have a parabola that just touches the
x-axis like that, just touches there, or just touches like
that, just touches at exactly one point, when b squared
minus 4ac is equal to 0. And then the last situation
is if b squared minus 4ac is less than 0. Then over here, you're going to
get a negative number under the radical. And we saw an example of
that in the last video. If we're dealing with real
numbers, we can't take a square root of a negative
number, so this means that we have no real solutions. In the future, you're going to
see that we will have complex solutions, but if we're dealing
with real numbers we have no real solution. Because this makes no sense. The square of a negative number,
at least it makes no sense in the real numbers. And then there's more
you can think about. If we do have a positive
discriminant, if b squared minus 4ac is positive, we can
think about whether the solutions are going to
be rational or not. If this is 2, then we're going
to have the square root of 2 in our answer, it's going to
be an irrational answer, or our solutions are going
to be irrational. If b squared minus 4ac is 16,
we know that's a perfect square, you take the square
root of a perfect square, we're going to have
a rational answer. Anyway, with all of that talk,
let's do some examples, because I think that's
what makes all of these ideas tangible. So let's say I have the equation
negative x squared plus 3x, minus 6
is equal to 0. And all I'm concerned about is
I just want to know a little bit about what kinds of
solutions this has. I don't want to necessarily
even solve for x. So if you're in a situation like
that, I can just look at the discriminant. I can just look at b
squared minus 4ac. So the discriminant here is
what? b squared is 9 minus 4, times a-- negative 1-- times
c, which is negative 6. So what is this equal to? This negative and that negative
cancel out, but we still have that negative
out there, so it's 9 minus 4, times 6. This is 9 minus 24, which
is less than 0. So we're going to have a number
smaller than 0 under the radical. So we have no real solutions. That was this scenario
right here. And so this graph is going to
point downwards, because we have a negative sign there,
so it probably looks like something like that. If that's the x-axis, the
graph is dipping down. Its vertex is below the x-axis
and it's downward-opening, so it never intersects
the x-axis. We have no real solutions. Let's do another one. Let's say I have-- I'll do this
one in pink-- let's say I have the equation, 5x squared
is equal to 6x. Well, let's put this in the
form that we're used to. So let's subtract 6x from both
sides, and we get 5x squared minus 6x is equal to 0. And let's calculate
the discriminant. So, we want to get b squared. b squared is negative
6 squared minus 4, times a, times c. Well, where is the c here? There is no c here. There's a plus 0 that I'm
not writing here. There's no c. So in this situation,
c is equal to 0. There is no c in
that equation. So times 0. So that all cancels out. Negative 6 squared
is positive 36. The discriminant is positive. You'd have a positive 36 under
the radical right there, so not only is it positive, it's
also a perfect square. So this tells me that I'm going
to have two solutions. So I'm going to have
two real solutions. And not only are they're going
to be real, but I also know they're going to be rational,
because I have the square root of 36. The square root of 36 is
positive or negative 6. I don't end up with an
irrational number here, so two real solutions that
are also rational. This is this scenario
right there. And you could also have
irrational in this scenario, so it's this [? here ?] plus the irrational. Let's do a couple more, just
to get really warmed. Let's say I have 41x squared
minus 31x, minus 52 is equal to 0. Once again, I just want to
think about what type of solution I might be
dealing with. So b squared minus 4ac. b squared. Negative the 31 squared minus
4, times a, times 41, times c-- times negative 52. So what do I have here? This is going to be a
positive 31 squared. The negative times
the negative, these are both positive. So I'm going to have
a positive, right? This is the same thing as 31
squared, plus-- this is a positive number right here, I
mean, we could calculate it, but it's 4 times 41, times 52. All I care about is my
discriminant is positive. It is greater than 0,
so that means I have two real solutions. And we could think about whether
this is some type of perfect square. I don't know. I'm not going to do it here. That would take a little
bit of computation. So we know they're real, we
don't know if they're rational or irrational solutions. Let's do one more of these. Let's say I have x squared
minus 8x, plus 16 is equal to 0. Once again, let's look
at the discriminant. b squared, that's negative 8
squared minus 4, times a, which is 1, times
c, which is 16. This is equal to 64 minus
64, which is equal to 0. So we only have one solution,
and by definition it's going to be rational. I mean, you could actually
look at it right here. It's x minus 4, times x
minus 4 is equal to 0. The one solution is x
equal to positive 4. And when I say by definition of
the quadratic formula, you look there, if this is a 0,
all you're left with is negative b over 2a, which is
definitely going to be rational, assuming you have a,
b, and c are, of course, rational numbers. Anyway, hopefully you
found that useful. It's a quick way. You don't have to go all the way
to solving the solution, you just want to have to say
what types of solutions or how many solutions, how many real
solutions, or inspect whether they're real or rational. The discriminant can be kind
of a useful shortcut. And I also think it makes you
kind of appreciate the parts of the quadratic formula
a little bit better.