Trending Math Olympiad question under Surds

Welcome back to my channel. Another Math Olympiad question. We are asked to find x if uh root x plus root 2 equals root 32. We're going to take it easy. I'm going to take it easy. So stick with it. You and I are going to enjoy the solution. You're going to enjoy the steps toward the solution. All right, now first of all we need to split this. Okay, we need to split this because root 32 It's the same thing as root 16 times root 2, right? Yes, root 32 is the same thing as root 16 times root 2. That is the first thing we need to do here. So we need to solve that root x. Plus root 2 is equal to root 16 times root 2. Okay, that is the first thing we need to do. So what do we do now? If we are to solve what we have here, that means we have √x plus √2. Is equal to the square root of 16, which is 4, right? 4 times root 2. All right, so we'll have root x. plus root 2 equals 4 times root 2, which is 4 root 2. We are approaching the solution. We are almost there. There. Okay, so what do we do next? Then we need to collect like terms. Yes, we have like terms here. Okay, so they're called similar swords. Similar. They're similar because we have root 2 here. This is the same as 1 root 2, and we have 4 root 2 here, so both of them are similar swords. So to bring them together, I need to explain something. When you have similar swords for example if you have 3 root 2 plus 4 root 2 this is like saying I have 3 oranges plus 4 oranges, which equals 7 oranges. 4 oranges, and when you have something like that, someone should say, "I have 3 oranges plus 4 oranges." The person should just say, "I have 7 oranges," right? So that means the same thing as 3 plus 4, which is 7 oranges, which is 7 root 2. So similarly, we are going to apply that here. Just see how we You're going to do it, but before that, if this is the first time you're coming across this channel, please don't hesitate to click that follow button. Just subscribe to the channel. Then help me by liking, commenting, and sharing so you don't miss my next video. All right. So what do we do from here? So we have that root x plus root 2 equals 4 root 2. So remember What I explained initially means that we are going to move this root 2 to the other side. so we are collecting like terms. Collect like terms. Okay, so that means the term with root x now will be 4 root 2, then this plus root 2; when you move this across the equation sign, it's going to be minus root 2. minus root 2. All right, so we now have that root x is equal to. So remember what I told you that The same thing as 1 root 2. So 4 is actually subtracting 1, which you have here. So 4 minus 1. will give us 3 root 2. Now remember that we are looking for x and not root x, and anytime you have Something like this—what do you do? To remove this square root sign, you need to take the square. of both sides. So we are going to square both sides. Square both sides. Okay, so you have root x squared is equal to 3 root 2 squared. Okay, now this square cancels with this root, and we are left with x equals remember that this square applies to both 3 and root 2. Okay, so 3 squared is going to Give us what, okay. Let's just do it. 3 raised to the power of 2, root 2. This dot means multiplication, please, or let me just write multiplied by square root of 2, squared, square root of 2, what root 2 squared, okay, let's continue, let's continue, okay, so what do we have, we have that x equals now 3 squared. 3 squared is what? Okay, sorry, let me start here. 3 squared is what? Nine times the square of root 2 is 2. Yes, let me explain—we all know that the square of root 2 is the same as root 2 times root 2, and according to the law of sod, we can bring this to under one word root sign so it's the same thing as root 2 times 2 which is the same thing as root Two times two is four, and the square root of four is two. So, the square root of two squared will give us two. That is why we have 9 times 2, which means that x is equal to 9 times 2. 18 and this is our answer.