 So, we've talked about doing a GCF, we're taking out the greatest common factor between multiple terms, bringing it out to the front, that way we can have two different things multiplied together to give us zero. So that way we can isolate, you know, take each one equal to zero, so we can solve the equation, right? There's going to be equations that we're going to get where there is no GCF, there is no way of taking out a GCF from all the terms to get two things multiplied together to give us zero, that way we can set each one equal to zero. So, in general, the equation is a quadratic equation, and we've talked about this before. The form of quadratic equation. Now this is the general form of quadratic equation. There's two different types of factoring that we're going to learn for this. One of them is where A is equal to one, and the other one is where A doesn't equal to one. It's going to be any other integer other than one, okay? Now the criteria for an equation to be quadratic equation, quadratic function, is if A does not equal to zero. So whenever you see something like this, where A does not equal zero, then you have a quadratic function. Now B could be equal to zero, it's still a quadratic function, C could be equal to zero, it would still be a quadratic function, and both B and C could be equal to zero, they would disappear, and you would still have a quadratic function, a quadratic equation where A x squared equals zero, or f of x equals A x squared. That's still a quadratic function if you only have this. So quadratic function, criteria for quadratic function is x squared. You could have an x term, or you could not have an x term, irrelevant. You need the x squared term in there, okay? What we're going to talk about is how to solve these quadratic equations using simple triangle. For example, we're going to have equations like, so we're going to have x squared plus 5x plus 6 is equal to zero, and we're going to learn how to solve these types of equations. Over here, this equation is in this form, the A just happens to be one. And whenever you have something like this where the A is one, that's a simple trinomial, and we're going to learn how to factor simple trinomials right now. And the method is quite simple. All you do is look for two numbers that multiply to give you six, and add to give you five, and you break it up into two different pieces, and you take the square root of this thing and put it up front. If we had something like this, there is no GCF that you can take out to have two things multiplied together, right? So we need to learn a new technique where we can break this thing out, break it up into two things multiplied together or multiple things multiplied together. So again, if they gave you something like this, they would just say factor this. But we're going to skip that part and we're just going to go straight to solve it, because it includes the factor in part. So all they would do, if it was given like this with no equal sign, they would just say factor. If they said equal to zero, then what you're doing is you're solving this equation. The way you tackle these types of problems is you're looking that basically it's a thought process, and you have to know your multiplication table for this. Super important. What you're doing is you're looking for two numbers that multiply to give you this, and add to give you this. And I'm just talking about the coefficients here, the coefficient here and the constant here, right? So what you're doing is, initially you can make up a table, but later on it's just going to become routine. The more of this you do, and you end up doing, most people in high school math, this is I think introducing grade 10 in where I am anyway. Well, it is introducing grade 10 where I am. And by the time you finish grade 10, you're going to end up doing hundreds of these. By the time you finish high school math, if you take all the way to grade 12, you're going to do so many of these, it's just going to become routine. And it's super important to know your multiplication table, because you always start off with the multiplication, because there are less numbers that multiply to give you six, less integers that multiply to give you six, than there are ones that add to give you five. What you're looking for is two numbers that multiply to give you six, and add to give you five. So what you can do is create a table, and what you do is, you say you're looking to multiply to give you six, and the sign in front of the number always goes with the number. So it's a positive six you're looking for. And they add to give you positive five. Now it should be fairly straightforward what the two numbers are, is two and six, two times, sorry, two and three, two times three gives you six, two plus five, two plus three gives you five. But let's start off with some of the other possibilities first, right? So what are two numbers that multiply to give you six, two integers that multiply to give you six? Because most of the problems we get, they're going to be dealing with integers. So two integers that multiply to give you six are going to be one times six is going to give you six, right? But one plus six is seven. It doesn't give us five, so that's not the number we're looking for. Another two number that multiply to give you six is negative one times negative six, right? Negative one times negative six, that's going to give you six. But negative one plus negative six, that's going to give you negative seven, and that's not what we're looking for. So basically what you end up doing in this column is listing all the possibilities until you find the combination you're looking for. You start listing all the possibilities, the numbers that multiply to give you six, and then you look over here to see if they add to give you five. The other two numbers, two other numbers that can multiply to give you six is negative two times negative three, and that gives you six. But negative two plus negative three gives you negative five, and that's not what we're looking for either, right? We're looking for positive five. So the two numbers that multiply to give you six, that add to give you five are two and three. So two times three gives you six, and two plus three gives you five. Now the way you factor this thing is, you want to break this up into two things, multiply it together to give you zero. So what you do is you go, right? For x squared, you have an x squared here term and an x term here. So all you do, you take the square root of the square root of x squared as an x, because you're going to break this up into two even segments, right? So over here, you got x and x, and you take your number, the two numbers that you found to give you this result, six and five, and they're both positive, so you're going to go plus two, plus three, okay? And this is this guy, Factor. Now if you want to solve for it, all you do is you take each one of these things, set them equal to zero. So this becomes x plus two is equal to zero, and x plus three is equal to zero. And you solve for these. You bring the x over here, it just becomes x is equal to negative two, right? Let's write this down here. x is equal to negative two, and over here, you bring the three over, which means x is equal to negative three. And these are your solutions to this question, okay? So you solve this equation. And again, you couldn't solve this using the simple rules that we learned of how to move around an equation, because you can't, there's no way of combining, you know, adding x squared and x, you can't do it. You have to break them up into two things, multiply together. Let's do a couple more complicated ones.