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Studying for a test? Prepare with these 8 lessons on Nombres complexes.
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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.