 So for example, let's do solve this equation here right now, right? So when you want to solve something like this where you have two x terms, they're different powers, so you can't combine them. What you want to do is take out a GCF from both of them. GCF meaning greatest common factor. So what's similar between both of these? Well, this contains only one x. This contains two x's. That's what x squared means. So you can take out an x from both of them. You couldn't take out an x squared because that doesn't have two x's. You can only take out basically a supply chain. I think what's it called? The weakest link in the chain, right? And this is the smallest link in the chain, so you can't take out anything more than this. So right now you can factor out an x. Okay. Now you took out an x from this. So basically what you're doing, you're going two x squared divided by x. One x kills one x. So you've got two x here, and x divided by x gives you one, right? And that's how you, you know, isolate this into one thing, multiply it to another thing to give you zero. And as we talked about before, if two things multiply them together to give you zero, then all you do is set each one equal to zero. So x is equal to zero, two x plus one is equal to zero. You're solving for both of them equaling zero because you don't know which one equals zero. So you're going to assume they both equal zero. So this is just a straight up answer, x equals zero. Over here, this becomes two x is equal to negative one divided by two divided by two. So x is equal to negative one over two. That's your second solution. And for those who already, you know, talked about quadratic equations, x equals zero and x equals negative one over two is where the parabola crosses the x-axis. So if you're looking at this, you would go x equals zero and x equals negative a half, if that's negative one, that's negative a half. That's where the parabola crosses the x-axis. And we know this parabola opens up because the number here is positive. So this is what our parabola is going to look like. We don't know how far down it goes. If you want to find out how far down it goes, all you do, you take the average of these two numbers, negative one over two plus zero divided by two is going to give you a negative a half, negative a quarter, and to find out the vertex of this parabola, all you do, you know, you already have the x value negative a quarter. And for the y value, you just plug in negative a quarter here and whatever that equals, that's your y value. And again, I've gone way too far here. This is just for those people who have already done quadratic equations that want to know what these terms represent. And these terms, whatever we're solving for something in mathematics, initially, right now, all we're doing is acquiring the tools that we need to be able to apply the information, right? Initially, when we learned how to add, we didn't go around calculating what something's going to cost us to buy or how much tax would be on something or what a discount payment or anything like this, right? All we did was learn how to add, right? This is the same thing. All we're learning right now is learning how to solve for a variable. And these are functions and these solutions right here mean something. Later on, we're going to look at what they mean, okay? Right now, it's really important for us to get this technique down to be able to do, you know, different types of problems or solve different types of functions that are to the power. What I've done so far is just taken it to the x squared term, right? What you can also do is have something like x to the power of five minus x cubed is equal to zero, right? Again, this is to the power of five, but this is two terms that we can solve for. So, what you do is you take a look at this and say, what's the greatest common factor that you can take out of those? That's x to the power of three, three x's there, five x's there. So, you can take out three x's from both of them, right? Again, you can't take out five x's because this doesn't have five x's. You go into the weakest link in the chain, right? And you're taking down an equation. So, this becomes x cubed comes out. What do you multiply x cubed by? You give it two x to the power of five, you multiply by x squared minus, you already have an x cubed. So, you have to have the spot, you reserve it by one, right? Whenever you do GCF, always remember, if you start off with two terms, you have to have two terms. A lot of people make a mistake when they say, oh, it took the x cubed out, so there's nothing here, it's just x cubed times x squared. Oops, I forgot the two right away. So, two, they forget to put the one. You have to reserve the spot. My basic, the, my analogy I use, I sort of seems to stick with people is, if you go to a movie theater with two people, right? If you go to a movie with your friend and your friend goes to the washroom, do you still keep their spot or do you give it away, right? You have to keep the spot. Your friend's still there. Just because they went to the washroom, they're going to come back at some point, right? So, always remember, if you start off with two terms or multiple terms and you're taking out a GCF, you always have to have the same terms of whatever the GCF is that you took out. And again, right now we have two things that multiply together to give you zero. You set each one equal to zero. So, x cubed is equal to zero and two x squared minus one is equal to zero. So, q root, you take the q root of this side to solve for x, right? The opposite of q. So, x is equal to zero. Oh, you're x squared equal to one. So, x, okay. So, the solution is going to be x is equal to square root of one over two. And that's partial answer. This is correct if you're in my part, in my area, in my part of the world, this is correct answer if you're in grade eight and nine. If you're in grade ten or above, the square root of anything is always plus or minus. So, this is actually plus or minus square root of two. As far as I'm concerned, that would be a correct answer if I was marking anything. If you only wrote down square root of, square root of a half, you wouldn't only get half marks. Square root of anything is always plus or minus because if you think about it, what you're looking for is, let's say you've got the square root of, let's say you've got the square root of four, right? The way it works is this. Let's say you've got the square root of four. What's the square root of four? It's not just two. It's plus or minus two. Square root means you need two identical things, the multiplier is going to give you this number. What are two identical things? The multiplier is going to give you four, right? It's two times two. Two times two gives you four, but you can also have negative two times negative two. Negative two times negative two gives you four. So, square root of four is not just two. It's negative two as well. Always has two solutions. Square root of anything. So, right now, for this question, we have one answer, x is equal to zero, x is equal to positive square root of a half, and x is equal to negative square root of a half, okay? And those would be our three solutions. And just for those people who are, who have already gone into these types of functions, polynomial functions and stuff like this, these points is where you cross the x-axis. And to the power of five, x to the power of five, this means your function has five curves, and three times it crosses the x-axis, and it crosses the x-axis at zero. Negative square root. Anyway, plus or minus square root of a half, right? If you weren't going to graph this, let's graph this real quick. So, what we would have is these are the solutions, x is equal to zero square root of a half, negative square root of a half, and you were solving for this equation, right? And those were your solutions, and those are really where the function crosses the x-axis. If this was a function where your zero is f of x, right? So, f of x is your y-axis, and this is your x-axis. x is equal to zero is here. x is equal to square root of a half is going to be here somewhere, and negative square root of a half is half is this, right? Now the way it works is this guy came to us from the cube root, x cube is equal to zero. The way this works is cube root, or x cube, when you're graphing a function it gives it a little curve. This is an odd power, positive in the front, so the function is going to look like this, okay? This part is going to go up, this part is going to go down, and your solution is where your function is going to cross the x-axis. So, it will look something like this, go up as wherever we go up to, all you would do is take the average of these and punch them in. It's not going to be exact when it comes to higher degrees, as it is with quadratic equations, but you're going to get an idea of where it goes, so this thing would come down. That was the cube root, so it's going to do a little twist. Again, it's going to do this, come back up, and up it goes, and that's what our function is going to look like, okay? Hopefully that's what it's going to look like.