 In this video, I want to solve an exponential equation, but one of the very different type than we saw previously. This equation is an example of what one often calls an equation of quadratic type. Or you could say it's like a quadratic-like equation. So what do you mean by that? What does it mean to be like quadratic-like? Well, the idea here is that a quadratic equation usually looks like something like ax squared plus bx plus c equals c, or it's something equivalent to that. But why not? It doesn't have to be the variable x, right? We could use any variable we want. So what if we call just, you know, let's pick a variable that we don't usually use. What if we call it like u, for example, a u squared plus b u plus c? Like I said, u is a variable we often don't use. We usually like to use things like x, y, and z. Also, it makes for some very interesting, grammatically correct, but awkward English statements when I say things like, oh, u is five. You know, that makes people go crazy. But a quadratic-like equation is gonna be the following idea that if you make the substitution, you're gonna make the substitution, let's say u equals f of x, and you put it into some equation, and you end up with something like the following. This is what we mean by a quadratic-like equation. And so what I claim here is that this equation you see on the screen is quadratic-like. But what's my u? It's gonna be a function of x right here. So I'm gonna make a substitution of the variables. Sometimes it's called the u-substitution, for which I'm gonna make the substitution u equals two to the x, okay? So if you do that, notice what you see is the following. We'll worry about the four to the x in just a second. But you see that negative two to the x, that just becomes a u. You have a minus 12, that's constant equals a zero. So how do you do with the four to the x? Well, this is the fun part here. When you take four to the x, four to the x by x one rule, notice that four is two squared. So this is really just two to the two to the x, which by x one of the rules, this is the same thing as two to the two x, like so. But on the other hand, if I were to take two to the x squared, this is equal to two to the two x. Notice the two things are equal to each other. So in general, right, this is a pretty neat little trick right here. If you have a to the x right here and you square it, this is the same thing as just a squared to the x, all right? So here we had two to the x as our u, then that means four to the x right here. Turns out that's just your u squared. If I took u to be three to the x, then u squared is gonna equal nine to the x. And so, you know, if u was four to the x, you'd end up with u squared being six to the x, something like that. And so that's a nice little recognition so that we can actually identify quadratic like equations when exponentials are involved very, very quickly. So coming back to our four to the x over here, we see that the four to the x is actually just u squared. And so with that perspective, you made this little u substitution over here. We see that, I'm gonna put a little different color so we can come back to it. This is something we wanna remember. By just changing the variables from x to u, you didn't see that this equation right here is essentially just a quadratic equation. We have a u squared minus u minus 12 equals zero. We can solve this like any other quadratic equation, which I'm just gonna do it by factors, factoring here. We need factors of negative 12 that have to be negative one. I can take u minus four and u plus three, right? Notice negative four times three is a negative 12, but negative four plus three is negative one. So we have the correct factorization there. And so then we end up with u equals four and negative three. Now, sometimes we get so excited about solving the quadratic equation. We forget that we made a u substitution here. We have to go back to the original variable, right? We have to go back to this u statement right here. So what this tells us is that if u equals four, that means two to the x is four. And if u equals negative three, that means two to the x equals negative three. Now on the left-hand side, there's a couple of things you could do. You could solve this by using a logarithm if you wanted to. This was not too hard to see, right? What power of two gives you four? Well, that's the second power. So that would tell me that x equals two because again, x squared equals four. On the right, the second one though, two to the x equals negative three. Well, remember the domain of an exponential function like two to the x is actually gonna be all positive numbers. Excuse me, not the domain, I meant to say the range. The domain is all real numbers. The range of two to the x, this is gonna be all positive numbers. So in fact, you can't have an exponential equal a negative when your input is a real number. So it turns out there's no solution that comes from this situation right here. So we can just get one solution and thus it turns out to be x equals two. And so we showed you how to show in this video how to solve a quadratic equation when your u substitution is an exponential function. But this idea works in general that if you can find u equals some f of x, right? And you make that substitution, you get this quadratic equation. Solve the quadratic equation, you're gonna get two solutions and then you have to solve the equations f of x equals the first solution and then f of x equals the second solution, which we did that with exponentials in this case.