Introduction to the quadratic equation

Welcome to the presentation on using the quadratic equation. So the quadratic equation, it sounds like something very complicated. And when you actually first see the quadratic equation, you'll say, well, not only does it sound like something complicated, but it is something complicated. But hopefully you'll see, over the course of this presentation, that it's actually not hard to use. And in a future presentation I'll actually show you how it was derived. So, in general, you've already learned how to factor a second degree equation. You've learned that if I had, say, x squared minus x, minus 6, equals 0. If I had this equation. x squared minus x minus x equals zero, that you could factor that as x minus 3 and x plus 2 equals 0. Which either means that x minus 3 equals 0 or x plus 2 equals 0. So x minus 3 equals 0 or x plus 2 equals 0. So, x equals 3 or negative 2. And, a graphical representation of this would be, if I had the function f of x is equal to x squared minus x minus 6. So this axis is the f of x axis. You might be more familiar with the y axis, and for the purpose of this type of problem, it doesn't matter. And this is the x axis. And if I were to graph this equation, x squared minus x, minus 6, it would look something like this. A bit like -- this is f of x equals minus 6. And the graph will kind of do something like this. Go up, it will keep going up in that direction. And know it goes through minus 6, because when x equals 0, f of x is equal to minus 6. So I know it goes through this point. And I know that when f of x is equal to 0, so f of x is equal to 0 along the x axis, right? Because this is 1. This is 0. This is negative 1. So this is where f of x is equal to 0, along this x axis, right? And we know it equals 0 at the points x is equal to 3 and x is equal to minus 2. That's actually what we solved here. Maybe when we were doing the factoring problems we didn't realize graphically what we were doing. But if we said that f of x is equal to this function, we're setting that equal to 0. So we're saying this function, when does this function equal 0? When is it equal to 0? Well, it's equal to 0 at these points, right? Because this is where f of x is equal to 0. And then what we were doing when we solved this by factoring is, we figured out, the x values that made f of x equal to 0, which is these two points. And, just a little terminology, these are also called the zeroes, or the roots, of f of x. Let's review that a little bit. So, if I had something like f of x is equal to x squared plus 4x plus 4, and I asked you, where are the zeroes, or the roots, of f of x. That's the same thing as saying, where does f of x interject intersect the x axis? And it intersects the x axis when f of x is equal to 0, right? If you think about the graph I had just drawn. So, let's say if f of x is equal to 0, then we could just say, 0 is equal to x squared plus 4x plus 4. And we know, we could just factor that, that's x plus 2 times x plus 2. And we know that it's equal to 0 at x equals minus 2. x equals minus 2. Well, that's a little -- x equals minus 2. So now, we know how to find the 0's when the the actual equation is easy to factor. But let's do a situation where the equation is actually not so easy to factor. Let's say we had f of x is equal to minus 10x squared minus 9x plus 1. Well, when I look at this, even if I were to divide it by 10 I would get some fractions here. And it's very hard to imagine factoring this quadratic. And that's what's actually called a quadratic equation, or this second degree polynomial. But let's set it -- So we're trying to solve this. Because we want to find out when it equals 0. Minus 10x squared minus 9x plus 1. We want to find out what x values make this equation equal to zero. And here we can use a tool called a quadratic equation. And now I'm going to give you one of the few things in math that's probably a good idea to memorize. The quadratic equation says that the roots of a quadratic are equal to -- and let's say that the quadratic equation is a x squared plus b x plus c equals 0. So, in this example, a is minus 10. b is minus 9, and c is 1. The formula is the roots x equals negative b plus or minus the square root of b squared minus 4 times a times c, all of that over 2a. I know that looks complicated, but the more you use it, you'll see it's actually not that bad. And this is a good idea to memorize. So let's apply the quadratic equation to this equation that we just wrote down. So, I just said -- and look, the a is just the coefficient on the x term, right? a is the coefficient on the x squared term. b is the coefficient on the x term, and c is the constant. So let's apply it tot this equation. What's b? Well, b is negative 9. We could see here. b is negative 9, a is negative 10. c is 1. Right? So if b is negative 9 -- so let's say, that's negative 9. Plus or minus the square root of negative 9 squared. Well, that's 81. Minus 4 times a. a is minus 10. Minus 10 times c, which is 1. I know this is messy, but hopefully you're understanding it. And all of that over 2 times a. Well, a is minus 10, so 2 times a is minus 20. So let's simplify that. Negative times negative 9, that's positive 9. Plus or minus the square root of 81. We have a negative 4 times a negative 10. This is a minus 10. I know it's very messy, I really apologize for that, times 1. So negative 4 times negative 10 is 40, positive 40. Positive 40. And then we have all of that over negative 20. Well, 81 plus 40 is 121. So this is 9 plus or minus the square root of 121 over minus 20. Square root of 121 is 11. So I'll go here. Hopefully you won't lose track of what I'm doing. So this is 9 plus or minus 11, over minus 20. And so if we said 9 plus 11 over minus 20, that is 9 plus 11 is 20, so this is 20 over minus 20. Which equals negative 1. So that's one root. That's 9 plus -- because this is plus or minus. And the other root would be 9 minus 11 over negative 20. Which equals minus 2 over minus 20. Which equals 1 over 10. So that's the other root. So if we were to graph this equation, we would see that it actually intersects the x axis. Or f of x equals 0 at the point x equals negative 1 and x equals 1/10. I'm going to do a lot more examples in part 2, because I think, if anything, I might have just confused you with this one. So, I'll see you in the part 2 of using the quadratic equation.