 In the previous video, we saw this quadratic formula, X equals negative B plus or minus the square root of B squared minus four AC all over two A. Now, we saw that by completing the square on a generic quadratic equation, we are able to solve every quadratic equation and we can do that with the quadratic formula. Now, this is a great tool we can use when we have to solve quadratic equations, but sometimes when it comes to solving quote unquote a quadratic equation, you don't actually need to solve it. Sometimes just knowing the nature of the solution set is good enough. So what does one mean by that? When you're trying to find out the nature of a solution set, sometimes it's good enough just to know like are my solutions gonna be whole numbers? Are they're going to be irrational, rational? Are they gonna be real complex? Do I have one solution, two solutions, no solutions? That's sometimes all that one needs to know when you're working with quadratic equations and the nature of the solution, it doesn't require the entire formula. It turns out that one only needs this part that's inside of the square root, this negative, or this B squared minus four AC part, which is referred to as the discriminant. So consider the following right here. So I mean, I just have it listed right here, this B squared minus four AC, this is what's referred to as the discriminant of the quadratic polynomial, for which we can see that when it's in standard form, AX squared plus BX plus C, it just takes the coefficients A, B, and C. And this discriminant can be useful to help us determine what's the nature of the solution set to said quadratic equation. So imagine we have a quadratic function, F of X equals AX squared plus BX plus C. And we then compute the discriminant of this quadratic function, we'll call it capital D. Now, first of all, consider that capital D is a positive number, B squared minus four AC. What that means is the square root of the discriminant is going to be a real number. And when you take the negative B in the quadratic formula, you have to add the square root of the discriminant, and you have to subtract the square root of the discriminant. And because of that, that'll give you two distinct solutions to the quadratic equation. Now be aware, if you have a quadratic function, F of X equals AX squared plus BX plus C, if you're solving the equation without equals zero, that means you're looking for the X intercepts. And so when your discriminant is positive, that indicates that you will have two distinct X intercepts, which of course are the solutions to the equation AX squared plus BX plus C equals zero. And these will be two distinct X intercepts and the equation, the quadratic equation will have two distinct real solutions. Now, when the discriminant is a perfect square, that means its square root will be a whole number. And that means that the solution to the equation AX squared plus BX plus C, the quadratic equation, will have two distinct rational solutions. That is to say the solutions will either be whole numbers or they'll be fractions. If the discriminant is not a perfect square, but it is still positive, that means you'll have two distinct irrational solutions to our equation AX squared plus BX plus C equals zero. And so whenever your discriminant is positive, you'll have two solutions to the equations, which will represent X intercepts on the graph. Perfect squares will give you rational numbers. Non-perfect square will give you irrational solutions. Now, another possibility is, what if your discriminant is equal to zero? B squared minus four AC equals zero in that situation. When we take the square root of zero, that is zero. And so that tells us that the solution to the quadratic equation will in fact be a rational number, but you only get one solution because if you add zero or minus zero, it does the same thing. In that situation, your solution would just be negative B over two A, like so. And you just give one solution to the equation AX squared plus BX plus C equals zero. Graphically speaking, that tells us that the graph would only have one X intercept. And what's interesting in this situation when you're discriminant is zero, this is actually the situation where AX squared plus BX plus C is already a perfect square. Because we got the quadratic formula by completing the square, it turns out if you're discriminant zero, the polynomial is already a perfect square and you could solve it by factoring, you just get the one solution. Now the third possibility is what if our discriminant is a negative number? If the discriminant's negative, you're gonna be taking the square root of a negative, which actually will give you non-real complex solutions. You'll get two different complex solutions to the equation AX squared plus BX plus C equals zero, and they'll be conjugates of each other, complex conjugates. You'll get some number that looks like X plus YI and X minus YI, something like that, two complex numbers, they'll be conjugates of each other. Now in terms of the graph, if we're graphing the function F of X equals AX squared plus BX plus C, if this equation has no real solutions, that actually means it'll have no X intercepts on the graph. Because the idea is, even though we can solve algebraically a equation for non-real solutions, when it comes to graphing, we only accept real numbers coming in and real numbers coming out. That was our original domain convention. And therefore there's no real number we could insert into the quadratic function that produces zero. So the graph will have no X intercepts to it whatsoever. And so I wanna illustrate again these three possibilities. The first possibility is when your discriminant was positive. When your discriminant's positive, you'll get two distinct X intercepts on the graph. You have two distinct real solutions, which will be either rational or irrational depending on whether D is a perfect square or not. If the discriminant is equal to zero, then you'll get one X intercept on the graph of the quadratic function. This will correspond to one real solution. And when your discriminant is negative, that means your quadratic equation has two non-real solutions. But in terms of the graph, the graph will have no X intercepts. So the graph is either entirely above or entirely below the X axis, as you can see in case number three right here. So what I wanna do then is just check, not actually solve the solutions of these things, but I just wanna check real quick what would the solution sets, what's the nature of the solutions for these quadratic equations. That's what we wanna check here. So for the first one, we have A equals one, B equals negative three, and C equals negative 40. So we wanna compute the discriminant. D equals, well, just remember what's inside of the quadratic formula there. You're gonna get a B squared minus four AC, right? So you're gonna get negative three squared minus four times one times 40. Negative 40, I should say. I miswrote that earlier. See, you have to be very careful. You grab the negative signs right here. So C is a negative 40. And so then continue on with this. We're gonna end up with negative three squared, which is a positive nine. Negative three times negative three is a positive nine. And then you're gonna get a positive because it's a double negative of 160. 160 plus nine is 169. And so 169, I wanna mention two things. It's 13 squared, but it's also a positive number. So what does this tell us about the solution set of this quadratic function? So what we're gonna see here is that since the discriminant is positive, we're gonna see that there are two real solutions to this equation. For which we could say there's also two X intercepts if we were to graph this thing. And but because the discriminant is a perfect square, we could also say the solutions are gonna be rational numbers. They're either whole numbers or they're fractions. One of those two things are gonna happen. We can continue to solve it if we want to. But if we just wanna determine the nature of the solution, well, we see that's gonna be two real, two rational solutions to this one. Well, how about the next one? The next one here says two X squared minus three X plus four equals zero. So we see that the A value is two, B is negative three and C is positive four. So the discriminant will look like B squared, which is a negative three squared again, minus four times A, which is two, times C, which is four. And then simplify this. We get nine minus four times four is 16. And if you double that, you get 32, for which we could stop right there if we wanted to. We already know that this is gonna be negative, but if you wanna be a little bit more precise, this is negative 23, but in particular, it's negative. And so what we see here is because the discriminant is negative, we're gonna see that the equation has two non-real solutions. Now, I wanna mention that it's not good enough just to say that it has two complex solutions because every real number is a complex number. It's like saying, oh, there are two plants growing in my backyard. What type of plants are we talking about? Are these bushes? Are these trees, right? So non-real solutions mean there's some non, there's imaginary component going on here. Complex numbers versus real numbers. It's not an either or situation. Real numbers are complex numbers. So because the discriminant's negative, we wanna say there's two non-real solutions to this situation. Of course, you could say there's no real solutions. That's another way of explaining this one here. Looking at the next example, we get 4x squared minus 12x plus nine. So you see that a equals four, b equals negative 12, and c equals nine. So the discriminant here, we're gonna get negative 12 squared, which is 144, minus four times nine, like so. Of course, the 144 is negative 12 squared. Four times nine is going to be, let's see, four times nine is 16 times that by, sorry, four times four is 16. Four times 16 times nine, that's gonna be 144. So this actually turns out to be zero. So what that tells us about our solution set is that because the discriminant is zero, so this equation has a single real solution. Turns out that this equation right here is a perfect square trinomial. In fact, we could have factored this thing as 2x. Let's see, there'd be 2x minus three quantity squared if we had factored it, and that would give you the solution of that one. And then for the last one, x squared plus 6x minus eight, we see that in this case, a equals one, b equals six, and c equals negative eight. So the discriminant, d, is gonna look like six squared, which is 36, minus four times one, which is four times, four times eight is 32. And so we see in this situation, you get 36. I'm sorry, it's gonna be plus 32. Watch out those signs there, because you have a negative eight. Let me go back and do this a little bit more detailed. We get six squared minus four times one times negative eight. This is 36 plus 32, which is gonna give us 68. 68 is positive, but it's not a perfect square. So what this tells us is that this equation has two real solutions, which are irrational, because the solutions are gonna involve this plus or minus the square root of 68.