Int Algebra: Predict Solution Type by Discriminant

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Quadratic Equations and Inequalities
Solving with The Quadratic Formula
Predicting Types of Solutions from the Discriminant Page [1 of 2]
Now in working through the solution for certain quadratic equations, we’ve seen a variety of things happening. We’ve
seen that we could have two different real roots. We saw that sometimes we could actually just have 1 real root and
sometimes we actually have no real roots, but instead two complex roots. This raises an interesting question. How
can you tell just by looking at the quadratic equation which situation we’re going to be in? Namely, are we going to
have two different real roots, are we going to have just one real root, or are we going to have two imaginary or two
complex roots? There’s three different possibilities and it turns out that information is encoded in the coefficients of
the equation. And let me show you how you can think about it. In fact, let’s put up the quadratic formula again and
now, where will we determine this thing? Well, take a look at the quadratic formula and what do you see? You see
that we have a -b plus or minus the square root of b2 minus 4ac.
Now where do the two solutions come from? The two solutions come from the plus or minus part, that square root
thing. So we can immediately see that if there’s plus or minus and the thing that follows the square root is 0, then, in
fact, there’s only going to be one solution. Right? So in fact, we can see that if that square root thing over there is 0,
then we have only one solution. How do we determine if it’s real or complex? Well, again, it’s the square root that’s
going to give us the answer. Do you see why? Because if that square root, the thing inside the square root is
positive, then we’re going to have two different real solutions. But if that thing under the square root is negative, then,
in fact, we’re going to have two complex solutions. So all we have to do is look at the thing under the radical. And, in
fact, that has a name because it’s so important. It’s called the discriminant. So the discriminant, which sometimes
people denote as d, is just the thing under the radical, which is b2 minus 4ac. And now, just thinking about it, this is
something you should definitely not memorize. We’re just thinking about it. If this number is positive, then what do we
have? We have two different real solutions. Then two real solutions. If d actually equals 0, then only one solution.
And, in fact, what is the one solution? You can see it. Look over there and put it as 0 under the square root. What is
it? It’s
2
b
a
?
. That’s what you see there. So, in fact, I’ll tell you exactly what the solution is. It equals
2
b
a
?
. Just as a
little side comment, there’s one solution and I can tell you exactly what it is. There’s no need to factor anything. It’s
just
2
b
a
?
. That’s it. And what if d were to be a negative number, if d is negative, then we have two complex solutions.
So, in fact, the discriminant has the power to tell us what situation we’re in. All you have to do is look at the
coefficients and make this thing up. b2 - 4ac. Again, don’t memorize it. Just think about it because it makes a lot of
sense.
Let’s try some examples together. I’ll use my example pen. Let’s try 2x2 - 5x - 7 = 0. And now the question is, don’t
solve it. Don’t actually solve it. Just tell me what kind of solutions are we going to have. Two different real solutions,
one real solution or two complex solutions? Well, all we have to do is compute the discriminates. And the
discriminant is, remember, b2 - 4ac. So it’s b2 - 52 which is 25, minus 4 times ac. So that’s going to be 4 x 14. And
what does that equal? Well, actually, I don’t even care what it equals. All I care about is that if it’s positive, negative
or 0. So let me see if I can actually compute this for you. Well, this going to be what? This is going to end in a 6 and
if I carry the 1, this should be a 56. And so what is this? Well, we can figure this out, I guess, too, if we have to. I
think this would be 30 minus 31, or maybe not. Let’s see. I wish I could do arithmetic a lot better. Oh, no. It should
be 29 I think or 56 or something or +56. Let’s see. I think that I’m making some major mistakes here. Let’s go back
and see if I’m computing this discriminant correctly or not. So let’s see. B2 - 4 x ac. But what is c? Where were you
when I needed you? c is negative 7, negative 7, not positive 7, but negative 7. You know, I thought we were working
together as a team here. If you're not going to help me, then I can’t help you. Now I didn’t mean to yell, but all right.
So this is actually -14. So really this is -14. This was a minor mistake. This is like one of those web clog up things.
You know, when the web screws up. Sort of freezes. I’m sorry about that. So my mistake. But it’s good you can
see these mistakes. So minus 14. So the minus times the minus makes it a plus which means now I get to add and if
I add those two numbers, now that I can do reasonably successfully. 5, 6, 7, this is going to be 81. Well, the only
thing I care about is its sign. Well, that’s positive and so what that means is what? This is going to have two, two,
that’s right. Two different real solutions. Now I wasn’t asked to find them. All I wanted to know was what kind are
there. So it turns out there’s two real solutions to this problem. Great.
Let’s try another one really fast. In fact, let’s throw this away over there. How about this one. 3x2 - 2x + 10 = 0. All I
want to know is what kind of solutions are there going to be? So let’s compute the discriminate. So what would it be?
Quadratic Equations and Inequalities
Solving with The Quadratic Formula
Predicting Types of Solutions from the Discriminant Page [2 of 2]
Negative b, well, that’s not negative b. That’s just part of the formula. What I want to look at is the discriminates.
That’s going to b2 - 4ac. B2 is going to be minus 22 which is 4, minus 4 times ac. Well, a is 3 and c is 10. So ac is 30.
So what does this equal? So d = 4 – 120, which equals minus 116. And that’s negative. So that means I’m going to
have two distinct solutions, however they’re complex solutions, they are imaginary solutions.
Let’s try one last one together. I’ll bet you can’t guess what the answer is going to be now. Let’s see. Let’s see.
Don’t be so smart. So x2 - 20x + 100 = 0. Let’s see what the discriminate would be here. It would be what? Well, it
would be, let’s see. The square root of b2 - 4ac so that’s b2 - 4ac so it would be this thing squared which would be 400
- 4 x ac. So that’s 4 times 100 and, yes. Your guess is probably correct. 400 - 400 = 0. So this is only going to have
one solution and, in fact, you can factor this pretty easily and see that this is actually a perfect square. So there’s only
going to be one solution to this. So the discriminant, it’s really great. The thing under the radical actually tells you
how many solutions you're going to have and what flavor they’re going to be. So really an important object down
there and I hope you enjoyed it. And by the way, don’t tell you friends about that mistake I made before, because, you
know, that’s just between us. I’m sorry I yelled at you.