 So, let's do one of our basic examples that we've been doing for simple trinomial factoring and complex trinomial factoring and show you that you get the same answer if you use the quadratic formula, right? So the question would be, they would ask you to factor this expression, right? Just to factor this thing. Now, if they said to solve it, they would have to be equal to zero here and we'll get there, right? So, what would be this factor that's, and we've already talked about this one, right? And two numbers are multiplied to give you six, I have to give you five, that's two and three. So if you factor this straight out, it becomes x plus two, x plus three, right? Now, we wouldn't use the quadratic formula to solve this, but right now we're going to do it just for the simple example, right? Because what it comes down to, the quadratic formula comes in real handy when you can't factor things manually, right? There's, you know, in real life, you're not going to get expressions like this, quadratic functions like this, where, you know, they're super easy to factor. In real life, you're going to get, you know, numbers that are, you know, decimals and fractions and they're extremely hard to factor manually. You can't factor them manually, right? You need to use the quadratic formula and that's where the quadratic formula comes in handy, right? So, but let's do it for the simpler stuff first and then we're going to do it for more complicated stuff, right? Things that we can't factor manually. So, the quadratic formula, and whenever you're using the quadratic formula, whenever you're writing an exam that, you know, factoring an exam or any exam where there's going to be factoring and solving equations involved, for sure, you should be memorizing this equation, but for sure, beside your name, wherever you, you know, write your name on a first piece of paper, whatever it is, write down this formula because you're going to use it a lot. If they haven't given it to you in a formula sheet, right? So, one advice I do have is, if you're going to write an exam, write down the quadratic formula on a piece of paper right beside you because you're constantly going to use it. So, our quadratic formula is x is equal to negative b plus or minus, or square root of b squared minus 4ac over 2a, right? a, b, and c. Those are our terms from the quadratic, from the expression that we have, right, from the quadratic equation that we have here. Well, that's not an expression. It's not an equation, it's an expression because there is no equal sign, right? So, what we'll do, we'll just put an equal sign equals to zero because that's where we're really going with this, right? And your a, b, and c is your a here, that's x squared. If there is no number in front of there, that means there is a 1 there, right? So, your a is 1, your b is 5, and your c is 6, okay? A good way not to get confused with this stuff is, whenever you're given something like this, you know, write a is equal to whatever, and b is equal to whatever, and c is equal to whatever. And whenever you're doing that, make sure that you follow the rule that the sign in front of the number always goes with the number, right? So, we've got a is equal to 1, b is equal to 5, and c is equal to 6. And what we're going to do is take a, b, and c and plug it into the quadratic formula, right? And we're going to do this on this side, I guess. Hopefully, I can write small enough to fit it all in, okay? So, what we got is, x is going to be negative 5 because the formula says it's negative b, right? So, our b was 5, so it becomes negative 5, plus or minus the square root of 5 squared minus 4 times 1 times 6, all divided by 2a, which is 2 times 1. So, all we do is just simplify that now. Whenever you're doing the quadratic formula, whenever you're using it, just put down x here, and just put equals further down, just going down. You don't always have to put the x value here, okay? So, what here is, negative 5 plus or minus square root of 25 minus 24, and that's just going to be 1. So, square root of 1 is just going to be 1, right? 25 minus 24 is 1, and the square root of 1 is just 1. So, it's going to be x is going to be equal to negative 5 plus or minus 1 divided by 2. So, what you got is x is equal to negative 5 plus 1 divided by 2, and x is equal to negative 5 minus 1 divided by 2. There's two answers here, right? So, negative 5 plus 1 is going to be negative 4 divided by 2 is going to be negative 2. So, one of your answers is going to be negative 2. Your second answer is going to be negative 5 minus 1, which is going to be negative 6. Negative 6 divided by 2 is negative 3. So, you have two answers. What this is, is x is equal to negative 2 and x is equal to negative 3. If you're talking about Cartesian coordinate system, the coordinates for that would be negative 2 and 0 and negative 3 and 0. Those would be your x-intercepts, right? So, if you're graphing this, just a quick teaser on where we're going to go with this, the coordinates, the x-intercepts are going to be negative 2 and 0. That's this guy there, and it's going to be negative 3 and 0, and that's going to be that guy there, right? Now, what we did was I changed our expression to an equation, and we used the quadratic formula, you know, your A, B and C, to plug it into the quadratic formula and solve it, right? And we got your coordinate system and your x-intercepts, right? Now, if this 0 wasn't here, if this wasn't here, the way we originally wrote it, this was an expression, and they wanted you to factor it, then what you would have to do is convert x equals negative 2 and x equals negative 3 to your brackets, to your factors. And the way you do that is in the following form. It's basically moving around the equal sign, right? So, what you would have is x is equal to negative 2 and x is equal to negative 3, and to write this as a factor in the factored form, you would grab the negative 2 and move it over, and you would grab the negative 3 and move it over. So, what you would do is grab your negative 2 and negative 3 and bring them over. So, what you have is x plus 2 is equal to 0, and x plus 3 is equal to 0, and those two expressions, multiplied together, give you the original expression, right? So, it's as simple as that for the quadratic formula. It freaks a lot of people out when they see this thing, because it's one of the larger formulas that you first get introduced to, right? But all it is, is just plugging in your A, B, and C from your quadratic functions, from your quadratic form expressions into the quadratic formula, and then crunching away, solving it and getting your factors, right? Getting your roots, getting your x-intercepts, getting your 0s, right? And if you want it just to be a factor of this thing, all you would do there is just move them over to the same side as the x, and as soon as you have 0 on the other side, right? All you do is just take the two things and multiply them together, or leave them the way they are, and those will be your factors. So, let's go do some more examples, and we're going to do a little bit more complicated. Obviously, you wouldn't use the quadratic formula to factor this thing, because we've already talked about it. It's a simple trinomial, and you can factor it as a simple trinomial. What I'm going to do as well is, I'm not going to write down the quadratic formula every time, right? Because we're running out of space, it's a small screen, right? So, remember this formula. It's one of the best things you can do for yourself when it comes to studying mathematics, is to memorize the quadratic formula, because you end up using it a lot. Because in general, you don't really get, in real life anyway, you don't really get quadratic functions, quadratic equations, where the numbers work out so nicely as negative 2 and negative 3, right? Usually, you're ending up with irrational numbers, you're ending up with root symbols in there, you're ending up with fractions, right? So, learn this formula. Super important, super important, okay? Let's go do some more complicated examples, and we'll do one of every single one of those, the way the discriminant works out. Right now, the discriminant is equal to 1. That's above 0, and the discriminant is greater than 0. That means it gave us two real groups, right? Two distinct real groups. So, we're going to do some more examples, where we only have one real group, where the discriminant is equal to 0. So, that way the parabola only touches the x-axis. We're going to do one where the discriminant is negative. That means the parabola doesn't cross the x-axis, okay? So, let's go do some more examples, and hopefully, you know, hopefully we'll practice this thing. It becomes as easy as it looks, I guess. I know it doesn't look easy, but it really is, okay?