 Welcome back. In this video, we're going to talk about the discriminants. This is actually a bit of a formula. This is actually a little piece of the quadratic formula. What this is basically used for is to figure out what kind of solutions you're going to have to a quadratic equation. Now, looking at the pictures that we have here, you can kind of tell, okay, I have, here's a parabola, here's another parabola, here's another parabola. These are the three different things that can happen. Whenever you solve quadratic equations, you're basically looking for zeros. Whenever you solve a quadratic equation or look for the zeros, you set the equation equal to zero and you basically solve. Now, when you set it equal to zero, this is what you're finding. You're finding this point here and this point here, where it crosses the x-axis, or this one point here, where it crosses the x-axis. Or, in fact, sometimes you can have complex solutions. You can have imaginary solutions where the parabola itself will not cross the x-axis. So this is kind of what the discriminant is. Now, your quadratic formula is x equals negative b plus or minus the big square root of b squared minus 4ac all over 2a. Okay, that's your quadratic formula. Notice the discriminant is just the b squared minus 4ac. This little piece in there is the only piece that you need to determine what your parabola is going to look like and how many solutions you're going to have. So just with this piece, if this piece is going to be greater than zero, basically if it's positive, if your discriminant here is positive, then you're going to have two solutions. Basically, your parabola is going to cross the x-axis two different times. Those are your normal run-of-the-mill type of problems you're going to see with solving quadratic equations. On the other hand, what can happen is that it's equal to zero, and then if it's equal to zero, you only have one solution right here. So that means the vertex of your parabola is actually connected to the x-axis. And then the third thing that can happen is that you have negative. So when you take the discriminant, when you plug in those numbers, the b squared minus 4ac, when you plug in your b's, your a's and c's, you plug them all in, you actually get a negative number. If you get a negative number, that tells you that well, with a negative underneath a radical, you get imaginary solutions, which is actually going to tell you that you're going to have two complex solutions. So there are no points that cross the x-axis here in the real, with the real numbers, but with the complex numbers, we actually do have solutions. That's kind of what the discriminant is used for. And again, this discriminant just comes from the quadratic formula. It's just the piece that is underneath the square root symbol. So a couple of examples of using this discriminant. What you can do is find the type and the number of solutions for an equation, even before you even solve it. So again, you can kind of get an idea if it's crossing the x-axis once or twice or if it's not crossing the x-axis at all. There's a couple different uses for this. But to do this first, what you need to do is you need to get everything on one side, negative 12x plus 36 equals 0. Take that negative 12x, subtract it over to the left side, and then you get this here. Now what this does, this tells you that my a number is 1, my b number is negative 12, and my c number is 36. So a is 1, b is negative 12, and c is a positive 36. Now what I'm going to do is I'm just going to plug that into the discriminant and then kind of interpret what that number is, and that'll tell me how many solutions I have. So in this case, my discriminant is b squared minus 4ac. That is just the piece that's underneath the square root of your quadratic formula. So b squared, so that's negative 12, parenthesis squared minus 4 times a, which is 1, and then c, which is 36. So in this case, 144 minus 4 times 36, 4 times 30 is 120, 4 times 30 is 120, 4 times 6 is 24, so that's going to be 144. Well, look at that coincidence. 144 minus 144 is 0. Okay, now I'm going to flip back my page a little bit. What I just figured out is that my discriminant is 0. So looking at my solutions right here is the picture that kind of matches up. This tells me that the parabola that I have from this function only hits the x-axis once. So find the type and the number of solutions. I have one real solution. I have one real solution to this. Now, this isn't something, I don't know, if you use this, this can actually tell you if you're going to have real solutions or not real solutions to your problem. This can have a couple of different uses depending on what kind of word problems you have or little things like that, but this is a really nice process to use just to figure out if you're going to have real solutions or imaginary solutions, real solutions or complex solutions. So to go over the next one, the next one's not really that different. 12x, excuse me, x squared minus 12x plus 40 is equal to 0. Now, noticing on this one got everything to the left side, set it equal to 0, and it's very, very similar to the first one, but notice that this one has a 36 and this one has a 40. So that right there is going to give us enough, if you remember what we did, it's going to give us enough to make this number down here and go through the process. My discriminant is b squared minus 4ac. My b number is negative 12. So square that minus 4 times a, which is 1, and times c, which is 40. The math here is a little bit easier. 12 squared is 144. 4 times 4 is 16, so this is 160. So this gives me a negative 16 for my answer. It gives me a negative 16 for my answer. Now again, if you remember back from that previous slide, if you remember back to that previous slide, I'll go back to it real quick. If I have negative, if it's negative underneath the radical there, then what happens is I have two complex solutions. This is where I start getting imaginary numbers. So I have two two complex solutions. There we go. There's my cursive s. All right. Anyway, that is, that's just using the discriminant. That's kind of what it's used for. There's not a whole lot of use for it outside of just your quadratic formulas. But it's just that little piece of the quadratic formula tells you so much. It tells you how many solutions. Are you going to have one solution or two solutions? Or are you going to have two imaginary solutions, two complex solutions? What are you going to have here? So it's actually a quite powerful little bit of the formula. Anyway, hope you enjoyed. Hope you enjoyed this video. And I think there's anything else I want to talk about. Anyway, hope you enjoyed this video and hope to see you next time.