A Math Olympiad exponential question/Germany/ how to solve for x

Welcome to my channel, and if you're a returning fan, take this hug. All right, we are asked to solve for x in this particular math Olympiad question. The first What you need to do is let's call this A, let's call this B. You know the deal now. So let's call this a, let's call this b, to make this look less ambiguous. So right, let's set a equals root x. Now, if we square both sides of this equation, we are going to get Let a squared be equal to x; let's call this Equation One. Similarly, let b equal the square root of what x minus three; if we should square both sides of this equation, we'll have that b squared equals x minus three, so let's call this equation two. All right, let's add. Add equation one to equation two. So, if we say one plus two. we will have that a squared plus b squared, right? a squared plus b squared is going to be equal to x plus x minus three x plus x minus three. All right, now, um, if we solve this further We will have that a squared plus b squared is equal to x plus x, which gives us two x, right? So we have two x minus three. Now let's call this equation three. If this is the first time If you're coming across this channel, please don't hesitate to click that subscription button. because trust me, you're going to be seeing this face all the time, right in front of you It's better if you click on that notification bell and also subscribe. Don't mind me, just click on it. the subscription button and on your notification bell and comment, especially if you have another way of solving this, thank you so much. All right, now what do we do next? We have from equation three. Now recall that we said let a equal this, and b equal this; that means we can say that a plus b, that is, a is this and b is that, is equal to three, all right? If we square both sides, squaring both sides of this equation, we will have that a plus b all squared is equal to three squared, okay? Um, using this, this is an algebraic identity, right? When we expand this, this is what we will have: a squared plus two a b plus b squared. is equal to three squared, which gives us nine; three squared gives us nine. Okay, now what do we do? Next, let's combine a plus b squared and move two ab to the other side of the equation. so we now have that a squared plus b squared equals nine, then this two a b we move to the other side to become minus 2ab. Now recall that we have a squared plus b squared in equation three. to be two x minus three, so we are going to substitute that here, so now say that two x minus three is equal to nine minus two a b, okay? Now recall that we said let's a equals Let a equal the square root of x and b equal the square root of x minus three; so let's substitute those into this. Okay, so we have two x minus three equals nine minus two brackets. Now our a is root x. right root x times our b is root x minus three, all right. Let's solve. So we have that two x Negative three is equal to nine minus two brackets; we can bring this under the same root sign, okay. We have square roots and square roots; we can bring them together, so we have x in brackets. Is x times what x minus three? All right, so this is two x minus three is equal to nine minus two. brackets square root of x times x will give you x squared x times minus three will give you minus three x. All right, now let's collect like terms. Let's collect like terms. Then we are going to have that, let's do x this two x here, let's bring this to the other side, so it's going to be two x minus we becomes plus two brackets square root of x squared minus three x okay is equal to this nine is already here, all right, then minus three crosses over to become plus three plus three, all right, so we now have, we can factor out a two from here since they have two as their common factor, so if we bring two out, we're going to have x plus what square root of x squared minus three x equals twelve; nine plus three gives us twelve. Now let's divide both sides of this equation by two to eliminate these twos. So, if we divide Divide this by two and this by two; this we can see, this and this we divide to give us six. So the new equation formed will be x plus the square root of x squared minus three x. Minus three x equals six. All right, just stick with me—you're going to enjoy the end of it. This particular question allows me to clean this side of the board, okay? Now we have that. Let's move this x to the other side, so we're going to have the square root of what x squared minus three x equals six minus x. All right, if we should square both sides of If we square both sides of this equation, we are going to have the square is going to cancel this square root right for this side and you are going to have x squared minus three x equals this, which is going to be six minus x squared. All right, now this is another algebraic identity: when we have something like this, we are going to expand it as follows: this x squared minus three x is equal to six squared, which is 36. then minus two times six is 12 times minus x, which gives you minus 12x then minus x squared will give you plus x squared, okay? Then let's continue, maybe I should just stand here, I feel better using my right hand here, okay? All right, now we have what x squared minus three x equals 36, okay, minus 12 x plus x squared, okay, this is what we Here we are now, what do we do next? Let's collect like terms. If we do that, we are going to have x squared this plus x squared we cross to become minus x squared, then we have minus three x here This minus 12 we cross over to become plus 12. x is equal to 36. All right, this minus this is zero. and 12x minus 3x will give us 9x, which is equal to 36, so let's divide both sides of this equation by 9 and we end up with x equal to 4. Now let's check. Let's check the equation; it says that square the square root of x plus the square root of x minus 3 equals 3, so we have the square root of 4 plus the square root of 4 minus 3, okay, is equal to what? 3. Now, the square root of 4 will give us 2 plus the square root of what? 4. -3 is 1 is equal to 3, so this is the same thing as 2 plus the square root of 1, which is 1, so 2 plus 1. is equal to 3, and that proves the solution. Thank you so much for watching, and do have a lovely day. Goodbye