 In this video, we provide the solution to question number 11 from practice exam number three for math 1050. We have the radical equation, the square root of x minus the square root of five minus x is equal to one. Our strategy when dealing with these square root equations is we want to isolate one of the square roots, then square both sides, and then iterate this process until all the square roots are gone. Because there's a negative in front of the square root of x minus five, I'm going to add the square root of x minus five to both sides, essentially moving it to the right hand side of the equation. This gives us the square root of x is equal to one plus the square root of x minus five. So this is the moment where we're then going to square both sides of the equation. Now be aware that when you square both sides of the equation, because the squaring function is not a one to one function, you're potentially adding fake solutions, these so-called party crashers we've talked about before. So we have to check our solutions at the end. Honestly, we should always check our solutions. But for this situation, even if we do everything right, if we don't check our solutions, we might find a solution that's not authentic, right? It's not a real solution. And I don't mean that as it's not a real number. It just it doesn't actually solve the equation. On the left hand side, when you square the square root of x, you'll get an x. On the right hand side, you do have to foil this thing out, right? Because it's one plus the square root times one plus the square root. So you get one times one, which is one. You'll get one times the square root, which is the square root of x minus five. Then you get the square root times one, so it's another square root of x minus five. And then you'll square the square root, so you end up with an x minus five, like so. The square root of x minus five times square of x minus five is x minus five. Let's combine some like terms, we have x is equal to, well you have one minus five, so that's a negative four, so you have x minus four there. And then you have two times the square root of x minus five. So I wanna move, again, I wanna get the square root by itself. So I'm gonna subtract x from both sides, that actually kills off all the x's. I'm gonna add four to both sides, add four. So then we end up with two times the square root of x minus five is equal to four. Since everything's even, I think I'll divide both sides by two as well. So divide by two, divide by two, the two's cancel on the left hand side. And of course, two goes into four, two times. So we have the square root of x minus five is equal to two. Then again, we're gonna square both sides, square both sides here. The fact that we might already have party crashers, we don't really acknowledge it again, since they've already entered, we know we have to check here. You square the square root of x minus five, you get x minus five. You square two, you're gonna end up with a four. Adding five to both sides, you get that x is equal to four plus five, which is nine. So it seems very impossible that since we only have one solution, that that's the real authentic solution. But one still has to be very cautious. So let's go back above and see what happens if we plug in nine. If you plug in nine here, you're gonna get the square root of nine minus four. Just looking on the left hand side. The square root of nine is three with the second radical nine minus five is four. So you get the square root of four, which is a two. Three minus two is in fact equal to one. This is a solution to the equation. Therefore, nine is an authentic solution. So we put a little box over, maybe put a bow, and it's good to go.