The most important thing you need to know to solve radical equations

What's up guys, in this video I want to start with the most important thing you need to know when you're trying to solve radical equations. So you can see in this example here, we have a 4 times the square root of 5x plus 1 minus 4 equals 12. Now, one of the most important things that we learn, or I want you to understand that is common amongst all functions, is this idea of what is inside the function and what is outside the function. So I want you to understand, this is extremely important for you to understand what we're doing. Under the square root, right, which is a square root function, you can see we have the 5x plus 2. On the outside, we have the 4 as well as the negative 4. So when you're solving square root functions, or really any kind of function, the main important thing that we want to do first is isolate the square root. We want this to be, you know, square root of whatever, you know, equals some number. Let's just call that C, okay? So we've got to get whatever is under the radical, we want that square root to be isolated so now we can go ahead and solve. So if you just think about this, and again, one of the things I like to do when we're first learning this is say, you know what, forget about the square root. What about if I just wrote this as a 4x minus 4 equals 12, right? What would you do? And again, guys, this is the relationship, like, what is actually x? X is actually a function, right? It's actually the identity function. So you can just kind of think about this and say, oh, if I was going to actually solve this two-step equation, first thing I'd do is add a 4 on both sides. Well, guys, look at this, it's the same thing, but instead of using the identity function, we're now using the square root function. Don't worry about what's inside the function, right? We're only focused on what is outside the function. So the first thing we're going to do is add a 4 to both sides. Now you can see over here, again, we'd have a 4x is equal to a 16. So then the next thing we're now going to do is say, all right, well, if I need to undo, right, multiplication by 4, I need to divide by 4. So you're going to divide using what we call the inverse operation, right? So that would be an x is equal to a 4. So over here, we need to undo multiplying by 4 by this radical function. So I'm going to divide by a 4 on both sides. And again, remember, this is 4 times the square root of 5x plus 1. So those are going to divide out just like those divide out. And now we're going to be left with a square root function or a square root equation where it is isolated. OK. So here's where it comes into the importance of understanding why we want to be able to have everything isolated into this form. So the reason is because we want to be able to get to using our inverse operations. We want to be able to undo the square root. And over here, we undid subtraction by adding a 4. Over here, we undid multiplication by dividing by 4. So one of the things that we got to make sure we understand is, well, how do we undo the taking the square root of something? And again, it's the same thing if you took the cube root or the fourth root. Like, how do we undo that? And the inverse operation of that is going to be squaring or raising to a power. Remember, we have an index here, which is 2, right? So if there's no number there, it's just assumed that is 2. So if I want to undo taking the square root, all I simply need to do is square both sides. Now when we're squaring both sides, we always need to be careful of creating extraneous solutions. We'll talk about that in some future videos. But for right now, I just want you to understand that the square root of something squared are inverse operations, just like they are did over here. And therefore, what they're going to do is now just going to give us an equation, which is 5x plus 1 is equal to 16. And now you can see that I can use my inverse operations to go ahead and solve this two-step equation. OK, so before I go ahead and finish the video, I want to make sure mentally, does this answer check? Does this answer work? And so when I plug in 3, 15, that's plus 1, 16. That's going to be squared. 16 is 4. 4 times 4 is 16. 6 times 6 is 12. And it looks like we are all set. Now this video is pretty straightforward. But if you want to know the next technique you absolutely need to know for solving radical equations, that's going to come up in the next video.