 So sometimes you might get a function, more complicated function embedded inside a quadratic function. For example, you might get something like this. So if we have something like this, five x squared plus five squared plus five bracket, x squared plus five plus three. Now what you gotta notice here is that is in the form of a quadratic function where x is equal to negative b plus or minus square to b square minus four is the over two a, right? I'm gonna stop saying this, but it's a good idea for you to continuously say it until it just rolls off your tongue, right? Like I said, you should have nightmares or dreams about this, right? So that's our quadratic function, but this x is not that x. This x is whatever is being squared, okay? And this x is when it comes to expressions where there is some confusion where it does confuse people. So what you should think about it is, again, this is not an x, it's whatever is being squared, which is in the form of the quadratic equation. So the way you should think about it is this guy here is representing our x here. So this is our box. This is our box, which is in the form of a quadratic equation and our box, not the x, is equal to the quadratic formula, right? That way, what you can do is use the quadratic formula to solve for this equation, right? Or to factor, this is not an equation because there's no equal sign there, to factor this expression, right? And at the end, what you're gonna end up having if this is factorable, you're gonna have x squared plus five is equal to this, right? So the way you should think about this, this box, x squared plus five, replaces the box. Now, you don't wanna sit there and write down x squared plus five is equal to your a, b's and c's in the quadratic formula, right? So b here is five, so this would be negative five, plus or minus the square root of five squared minus four, our a is two, and our b, and our c is three, divided by two times a, which is two times two, right? So that's what the quadratic equation gives us, right? But over here, you don't wanna constantly write down x squared plus five, right? What you wanna write down is just one letter, one expression, one term, so that way you don't have to worry about it. And so the way we do this is, what we do is use the let statement. So what you can say is let w, so we're gonna say let w equal x squared plus five, and that gives us a certain amount of control to rewrite that function or that equation or that expression in the form of a quadratic equation that we can recognize instantaneously. So as soon as we say let w equal x squared plus five, all of a sudden we're gonna have w is equal to this, and our above equation turns into the quadratic formula, which is what we want, right? So it turns into, let's write this guy here, if we say w, let w equal x squared plus five, so what this turns out to be is, right? Hopefully you can see that. So at the beginning here, the way we're gonna do one in order, we're just sort of trying to, I'm trying to go through the explanation of, you know, the mental process and why we do this, is just to make life easier for us, right? So what we end up having here is gonna be w is equal to, you know, this guy's simplified. So that's gonna be negative five plus or minus 25. Four times two is eight, eight times three is 24. Minus 24 divided by four, right? 25 minus 24 is more, so the square root of one is just gonna be one. So it's gonna be negative five plus or minus one over four. W is gonna be negative five plus one is gonna be negative four divided by four is gonna be negative one. And four, if you wanna think about it, it's an or, but it's an inclusive or, okay? So it's this or that or both, okay? So this negative, so negative five plus one was negative four divided by four is negative one. Negative five minus one is negative six divided by four is negative six over four. And that can't be simplified into negative three over two. So right now we have two answers, right? We have w is equal to negative one and we have w is equal to negative three over two. Now we haven't finished solving this equation yet because in this expression, if you're given that question to factor that question or to solve that question, what you need to do is it would basically be solved, right? They wouldn't give it to you to say factor. What we're working towards is equal to zero and we're trying to solve this equation, right? So if you got w is equal to negative one, w, you gotta have to go back to the original where you let statement when you began this expression began solving this equation. W is equal to x squared plus five. So what you end up having is right down here, you're gonna substitute w is equal to x squared plus five for w and then solve for x because x is what you're looking for. So what we had was we let w equal x squared plus five and we solve for this and we got w is equal to negative one and w is equal to negative three over two, right? I'm gonna write these a little bit bigger so they come up nicer. So what we end up doing is substituting this for w. So what we're gonna end up having is x squared plus five is equal to negative one and x squared plus five is equal to negative three over two. Okay. Now, right now, you're gonna have to solve for x squared. So what you're gonna do is move the five over. So you're gonna have x squared is equal to negative six, square root both sides and x squared is gonna be, or x is gonna be square root of negative six, right? And you can't take the square root of a negative number so this one doesn't give you an answer. There is no solution for x here. Over here, we're gonna have the same problem. If you don't like fractions, get rid of your fractions right away by multiplying by two, right? If you can multiply the whole equation by two, so you're gonna have two x squared plus 10 is equal to negative three and again, we've talked a lot about this in series three, and you should know how to do this if you don't already. And then grab the 10 over, so you're gonna have two x squared is equal to negative 13 divide by two, divide by two. So x squared is equal to negative 13 over two. And then if you take the square root of both sides, you're gonna have x is equal to square root of negative 13 over two. And again, this one is not gonna give you a solution. Okay. So the basic format is, if you get something that looks a little confusing, what you can do is use the let's statements to make substitutions into the expression or the function or the equation that they've given you for you to be able to put it in the form that you can recognize that you can use the quadratic formula in, okay? Let's do one where we did as a complex trinomial factoring and use the let's statement and use the quadratic formula just to prove to yourself that we're gonna get the same answers, okay?