 But my perceptions are more and more in the maybe mode. It's beginning to come out of my frontal lobes, into my back brain. I'm beginning to more and more perceive things as maybes. Although, of course, the only thing about maybe logic, your critics never noticed that. I put things in maybe terms, and I noticed almost all the really negative criticism I get is by people who assume, because I'm not taking their stand. I must be taking the opposite stand. They have a yes-no logic. I put things in maybe. They assume that means I put it in either yes or no. They can't understand maybe at all. Hi. This is Cheecho again. And we're just going to continue with the section that we're on right now, which is a factoring section. But we're going to do a little diversion and just talk about a certain concept in mathematics that I don't think is taught very well in general. Or it was confusing for me when I was, well, I don't know if I was taught it or it was just assumed that I'm supposed to know this. But it's basically the concept of a let statement or substitution. And in mathematics, again, it's just a language, right? And we can manipulate language any way we want to get an idea across or to make a certain concept easier to understand. And that's what the let statement is really about. Or that's what substitutions can be about, I guess. That's the best way of putting it. What happens is we're given equations for specifically when doing factoring polynomials. We talked about prime factorization, that's series one. But right now we're sort of going off on factoring polynomials. And the reason we're doing this is basically because we want to break down our functions to what they're made out of. Because once we take a function and we break it down into score elements, then we can look at the function and try to figure out what parts of it are common between other functions, right? What are its building blocks, right? That's why we're doing factoring. Now, so far we've been going off on, we've learned the simple trinomial, complex trinomial, difference of squares in GCF, right? And we started talking about the quadratic formula. And the quadratic formula and synthetic division, these two factoring techniques are gonna give us a lot more power because we can go beyond just simple polynomial functions, simple quadratic functions or simple trinomial functions, right? We can start actually factoring or breaking down functions that we can't easily break down into integers and multiplies together, which is the case that's going to be in real life, right? So the quadratic formula and synthetic division are gonna give us a lot more power. And what's gonna happen with this power that we have, we're gonna start going into functions, equations that are way more complicated, right? And what's gonna happen is we're gonna look at these functions, we're gonna encounter functions that may not look like we can do anything with them because they're not the general form that we're used to seeing them in, right? So for example, for the quadratic, we talked about the quadratic formula and the quadratic formula is used for quadratic equations, right? To solve, to break down quadratic equations. And the terminology we've used so far is the following. So so far we've got, basically, we've learned about the quadratic formula and we've done like three examples like this, where the discriminant is graded in zero equal to zero and less than zero, where grade in zero gives you two x-intercepts, equal to zero gives you one x-intercept and when the discriminant is less than zero, you got no x-intercepts, right? Now, the terminology that you're basically gonna encounter is, you know, if you have a quadratic equation like this, a x squared plus b x plus c, you can factor this out as x is equal to negative b plus or minus the square root of b squared minus four, it's the over two a, right? Now, this is true, but for me, this was initially when I learned this was a little confusing because this x and this x are, you know, for this expression, they're the same x, but what happens is you can get functions where it's not a x squared, but the x here is to a higher power, is to the power of four or six or eight and all you need to be able to use the quadratic formula and to use simple trinomial factor and complex trinomial factor is this power to be twice as much as this power, right? Following the properties of a polynomial where the properties of a polynomial that we talked about in series 3A with, you know, video number 89, 90, and 91 where the powers here must be natural numbers, whole numbers, there's an x to the power of zero here, but we avoid that, right? Because anything to the power of zero is just one, right? So as long as the power on the x's is natural and this is natural number and this power is twice as that power the quadratic formula applies. And when I first learned how to use the quadratic formula, this confused me because every now and then we would get an equation where, you know, the questions that I would get is to factor something that wasn't just x squared but x to the power of four, x to the power of six, x to the power of eight. So every now and then you would get equations like this. So you get ax to the power of four plus bx squared plus c. Now what happens in this case? The quadratic formula still applies but it's no longer x that equals this but this becomes x squared because the quadratic formula works for functions for equations that are in this form and this guy is not in this form. So what we have to do is convert this to something that's in this form, right? And knowing our, you know, exponent rules, our radical rules, where we have something like this we can break this down, right? x to the power of four is just x squared to the power of two. So we can break this equation down to the following form. Now if you take a look at this, this says ax squared to the power of two plus bx squared. And what I'm gonna do is gonna put the bx squared in brackets, right? Plus c. Now what we have here is this equation is in this form where the x here is the same as the x squared here, right? And this is where the confusion came with me when I first learned this was because right now if we have this equation in this form, the quadratic formula is not x is equal to this anymore. It's x squared is equal to this. So the quadratic equation changes, it becomes. So it becomes x squared is equal to negative b plus or minus a squared of b squared minus four is c over two a, right? So once you get an equation like this all you have to do is realize that this expression here, this definition isn't what x is equal to. It's not just what this expression here is just not what x is equal to. It's what x represents and x here represents anything which is basically our boxes, right? When I started for the introduction for the quadratic formula, you know I wrote down the equation, one form of the equation where I said we're gonna talk about it further and this is really what we're doing right now. We're sort of expanding on that concept. And I wrote down the general function in the following form. Where I said your quadratic equation you could use for anything written in this form where it's a x to the power of two n plus b x n plus c where n is an element of the natural numbers, right? But it goes beyond this because what happens is you can think about the quadratic equation as b, a box. So what you would have is a box squared plus b box plus c. If you have anything in this form you can write that as, let's shade the sense so it's more clear. So a box squared plus b box plus c is equal. If you wanna factor that it becomes box is equal to the quadratic formula. And remember we talked about this with the difference of squares, a box or a square. I like calling it a box because you can put things in a box, right? So the box represents anything you can put in, right? So anything, so a anything squared plus b anything plus c you can factor out in this form, okay? And this becomes super handy because now what happens is we can factor quadratic or equations like this and this is no longer a quadratic equation or you could write it as a quadratic equation when you rewrite it, right? Now what happens in mathematics when we start doing higher level mathematics we're not gonna use boxes, right? That's just, it's cute and all and it's a great way to introduce the concept but the best way to work with this is not to put boxes here but to put a different letter. So what we could end up doing is saying if let w equal x squared, right? And if we write it that way what we end up having is let's say in here you wanted to factor this guy what you could say is let w equal x squared, right? So if you let w equal x squared and that's the power of a language, right? You can say anything you want as long as you stay consistent with whatever you're saying. So what you can do is break things down even further or condense things that way you can write expressions like this in a form that you can actually understand, right? So as soon as we say let w equal x squared well, x to the power of four if this is x to the power of four x to the power of four is x squared, squared x to the power of two to the power of two, right? But we just said let w equal x squared so we can write this equation as we can write this equation as a w squared plus bw plus c. Now if we do it that way then right away we notice that this is a quadratic equation even though inside of it is embedded another function, right? Because this right now becomes a function w equals x squared. That's the same thing as like saying y is equal to x squared but we're not gonna use y or x because y and x are really generic. We use those terms a lot, those letters a lot to represent a Cartesian coordinate system and unknowns. So what we start doing is we start introducing other letters in the alphabet and you can use any letter you want, right? Preferably not a, b and c because they're already part of the quadratic formula, right? And you wouldn't use x and you generally don't use y because this if you're gonna set it as a function you're gonna set this equal to y or f of x, right? So you're not gonna use y or f of x. You could use k, you could use h, you could use w, you could use z. You can use almost any letter you want to create a new function and that way you can substitute that into an expression that may not look like a quadratic formula and rewrite it so it looks like a quadratic formula and that way you can apply the quadratic formula, right? Which is a lot better than saying x squared is equal to the quadratic formula here because sometimes what you end up having inside the box is a much more complicated function.