Harvard University interview math question|Algebra| can you pass?

Let's solve for the value of x. First, we need to move this term to the other side of the equation. so when we do that we have the square root of 2x equal to x minus the square root of x. now let's square both sides. sides of this equation, when we do that, we are going to have the square root of 2x squared is equal to x minus the square root of x squared; right now, the square is going to remove this root square root So we have 2x is equal to. Now I want us to recall that's where you have a when you have a minus b squared, okay, this is equal to a squared minus 2ab plus b squared, okay, now in this case Our a is what x, while our b is root x, so we can now write here that this is equal to 2x. That is, 2x equals then a squared, which is x squared, right, minus 2ab, which is 2. multiplied by x multiplied by the square root of x, okay, then plus b squared, that is plus the square root of x squared All right, okay then. Let's go ahead and solve that. This is sorry, I'm being confused by this. That means 2x is equal to x squared minus 2 times x, which gives us 2x root x. Right, plus this, you remove this, and we have x remaining, so plus x. Okay, so um, let's collect. So let's transfer this to the other side and solve; so we have x squared minus 2x root x. plus x, okay, then this 2x we cross over to become minus 2x is equal to zero, so we have x squared minus 2x root x now plus x minus 2x gives us minus x, right? Is equal to zero, okay now. I want to transfer this minus. Okay, looking at this, we can factor out x. Let's factor out x. so that we can actually simplify this equation, so when we factor out x, x squared divided by x gives us x minus now 2x root x divided by x gives us 2 root x, right now minus x divided by x gives us minus one, which is equal to zero. Now at this point, you can see that we have Here we have two equations: x equals zero, and x minus 2 root x. minus one is also equal to zero, okay. All right, now this is what we have. So from here, we have the first value of our x, which means that x is equal to zero; now, for this side, let's now Focus on this part, which says that for this one, which says that x minus 2 root x minus 1 equals zero, so we have that minus 2 root x equals, let's transfer this minus one to this side to give us one and transfer this x to this side to give us minus x all right Okay, now we can write that this is what we have here. All right, let me recopy it here, so we have minus 2 root x equal to 1 minus x, that is exactly what we have there now, let's divide both sides by minus 2 divided by minus 2, so we can see that root x is equal to 1 minus x divided by 2. Now let's square both sides of this equation. When we do that, we have this squared and this squared, okay, so we remove this, we have that x is equal to, so this is what we have here 1 minus x squared divided by 2 squared, all right? Okay, now recall that we have expand something similar here that (a minus b) squared is equal to a squared minus 2ab plus b squared; in this case, our a is going to be 1, while our b is going to be x, all right So we can now write that x is equal to, so it's going to be a squared, that is 1 squared. minus 2ab, which is 2 multiplied by 1 multiplied by x, plus b squared, which is plus x squared divided by 4, okay, so we are going to have that x is equal to 1 squared, which will give us 1 minus 2x right plus x squared divided by 4 okay now at this point we need to cross multiply This is actually x divided by 1, so let's cross multiply. When we cross multiply, we have 4x is equal to 1 times the entire expression; this will still give us the same result, but let me rearrange it. this is very good, so let's write x squared first, then minus 2x, then plus 1, right? Okay, now if this If this is the first time you're coming across us, please give us a thumbs up and turn on your notification bell. and also subscribe to this channel, okay? Thank you so much. Now, what do we do next? I want us to set this equal to zero, okay? So let's set this equal to zero. So we have 4x minus x squared; that is, when this crosses over, becomes minus x squared, then we have plus 2x. That is, when this crosses over, it becomes plus, and when this crosses over, it becomes minus one. is equal to zero, okay. All right, now so this is what we have here, this is what we have here. So we have that 4x, okay, sorry, I don't want us to confuse this with the other one, so this how we stop that 4x minus x squared plus 2x minus one is equal to zero, okay, so what we need to do Now let's actually solve this: 4x plus 2x will give us this x, then we have minus x squared. minus one is equal to zero. Now let us rearrange this. Well, so this is a quadratic equation, so our x squared is supposed to come first, okay? So we are going to write minus x squared plus 6x minus one. is equal to zero now because of this minus sign here. Let's multiply through by what? Let's multiply times minus one, so when we do that we have x squared minus 6x plus one equals zero now we cannot solve this using the factorization method, so what we are going to do is that we are going to use the quadratic equation formula, which states that x is equal to minus b plus or minus the square root of b squared minus 4ac, divided by 2a. Okay, so we are going to note that our a in This case is equal to the coefficient of our x squared, which is one; our b is equal to the the coefficient of x squared is one, and b is the coefficient of x is minus six, so we have minus six here, and our c is our constant, which is one. All right, we are going to substitute this into this particular equation, so that means that x is equal to minus b that is minus minus six will give us plus six plus or minus the square root of b squared that is minus six squared, which will give us 36, right? Minus 4ac, that is four times one times one, which will still give us one, all right, divided by two times a, a is one, which is still two, so that means that our x is equal to six plus or minus the square root of 32, all divided by two. Now we I can see our 32. Let's try to solve 32. If we solve 32, when 2 divides 32, we are going to have um 16. so we have a perfect square and also a regular number, therefore we can say that this implies that x is equal to 6 plus or minus the square root of 16 multiplied by the square that's 16 times, we still get 32 divided by 2, okay, so we continue from here, we can now write That our x is equal to 6 plus or minus the square root of 16, which gives us 4, we have the square root of 2, right? divided by 2, so this implies that x is equal to 6. All right, let's factor out 2 from here. if we factor out 2, so we have 2 here, okay, we open our bracket; now, when 6 is divided by 2, we have 3 plus or minus when 4 root 2 is divided by 2 we have 2 root 2 okay all right so we close our bracket divided by 2; now this, we divide this, therefore the second value of x will be 3 plus or let's bring out our x first: 3 plus 2 root 2 plus or minus; therefore, the second value of x It should be 3 plus 2 root 2, while the third value of our x is 3 minus 2 root 2. Thank you so much. For watching, have a lovely day, bye