A crazy maths Olympiad algebra question that almost all failed|can you solve for x?

Hi, let's quickly solve for the value of x. Um, first of all, I want us to bring this minus 4. to this side of the equation, okay, and if we do that, we have x minus 10 divided by x minus 2 okay plus x minus 9 divided by x minus 3 plus x minus 8 divided by x minus 4 plus x minus 7 divided by x minus 5 plus 4 is equal to zero I'd like you to watch to the end I'm going to explain this in detail so you definitely understand it, okay. Alright, let's continue. Now note that 4 is the same as 1 plus 1 plus 1 plus 1. So I can distribute those to each of these terms so I can write x minus 10 divided by x. -2 + 1, all right, + x - 9 divided by x - 3 + 1 + x - 8 divided by x -4 plus 1, okay, then plus x minus 7 divided by x minus 5 plus 1 is equal to zero, all right Now note, 1 plus 1 plus 1 plus 1 is already 4, so we haven't changed anything. Now let's all Okay, this first one here, but x minus 10 divided by x minus 2 plus 1 over 1. Right now, if we solve Ranger, this is equal to the LCM, which is平均 x minus 2, right? Okay, now x minus 2 divides x minus 2. have 1 times x minus 10, we still get x minus 10; plus, if 1 divides x minus 2, we still get have x minus 2 times 1, which gives us x minus 2. Okay, now this is equal to, if we solve this, we have x plus x gives us 2x, and minus 10 minus 2 gives us minus 12, divided by x minus 2, okay. Now, similarly, okay, if we should solve the second one, which is this: we have x minus 9. over x minus 3 plus 1, the LCM is x minus 3; when this divides this, we have 1. 1 times this will Give us x minus 9 plus this is 1 over 1, right? When 1 divides this, we have x minus 3 times 1. We still have x minus 3. Now, if we solve this, we have x plus x, which gives us 2x minus 9 minus 3. We give ourselves minus 12 divided by x minus 3, and also, if we should solve the third one, which is this. x minus 8 divided by x minus 4 plus 1. Okay, now the least common multiple is what? x minus 4. When this divides this. We have 1 times this, which is x minus 8, plus when this divides this, we have x minus 4 times 1. we still have x minus 4, so this is equal to x plus x, which gives us 2x minus 8 minus 4, which gives us minus 12 divided by x minus 4. Now I want us to check some things, like we have a trend, right? Check the notice that in all of them, the numerator is 2x minus 12, 2x minus 12 over this denominator. so this one is 2x minus 12 over x minus 3; this one is 2x minus 12 over x minus 4; so notice that if we solve this we are going to have 2x minus 12 divided by x minus 5, okay, so let's not waste our time solving it, so we are now going to substitute the entire thing into what we just Got it, so this one will be 2x, let me cover this, so this one will now be For this particular one, it will be 2x minus 12, okay? Oh, you can't see that? Sorry, so 2x minus 12. divided by x minus 2, right? Plus, now for the second one, which is this: 2x minus 12. 2x minus 12 divided by x minus 3 plus for the third one, which is also the same thing This one is still the same thing, that is 2x minus, and then I want, I want you to be able to see. what I'm showing you, yeah, so this third one now will still be 2x minus minus 12 divided by x minus 4 Okay, so that's what I will write: 2x minus 12 divided by x minus 4, plus then the last one. will be 2x minus 12 divided by x minus 5, so we have 2x minus 12 divided by x minus 5, okay, and this is equal to zero, all right. Now that they have the same thing in the denominator, we can Actually, factor that out, okay? So let's go ahead and do that. So we have 2x minus 12 in brackets. now when this is divided by this one, we are left with as the numerator, so it's going to be 1 over x minus 2 plus the same thing here, 1 over x minus 3 plus the same thing here, 1 over x minus 4 plus the same thing here 1 over x minus 5, all right, and this is equal to zero, all right now I want us to look at this: we have minus 2, minus 3, minus 4, minus 5, but if I combine x minus 2 and x minus 5, I will get the same numerator when I combine x minus 3 and x minus 4, since minus 3 Minus 4 is minus 7, and minus 2 minus 5 is minus 7, so we're going to use that as an advantage, okay. so we have um what I'm not going to add this to this one right 1 over x minus 2 plus 1 over x minus 5; now the LCN is (x minus 2) multiplied by (x minus 5). Right now, when this is divided by this Sorry, when this is divided by this, we have a remainder of x minus 5; x minus 5 times 1 will still Give us x minus 5, okay, plus oh, sorry, what am I doing? When this divides this, we have x minus 2. so x minus 2 times 1 will still give us x minus 2, okay. All right, now this is equal to So we have x minus 2 times x minus 5. Okay, now x plus x gives us 2x, and minus 5 minus 2. Give us minus 7, okay? Now let's do the same here. We are going to combine this and this to get the same thing as the numerator, so 1 over x minus 3 plus 1 over x minus 4; the LCM is x minus 3 brackets x minus 4, when this device, we have this, this times 1 is x minus 4, right, plus when this device We have this, this times 1 is 2x minus 3, so when you solve x plus x will give us 2x minus 4 minus 3 minus 7 times (x minus 3) times (x minus 4). All right, now we have the same numerator. You can see that, right? So that simply means we can substitute these two values here, okay? so what we now have is what 2x; so what we now have is what 2x minus 12 open brackets oh you I can see the open brackets, so we are going to add this to this because it's everything we have. Here we are going to add them both, so 2x minus 7 divided by x minus 2. (x - 5) plus then this one (2x - 7) divided by (x - 3)(x - 4) You know, I promise you I'm going to make you understand this, and I think I'm doing justice to that, right? Okay, now let's continue. We have the same numerator here, so we can still factor it out. so now we have 2x minus 12 multiplied by 2x minus 7 open brackets 1 over x minus 2 brackets x minus 5 plus 1 over x minus 3, closed bracket, x minus 4, okay, and this is equal to 0. Now I want us to look at this very well, at this point we can now equate that 2x minus 12 is equal to 0, right? Yes, 2x minus 7 equals 0, right? Then the whole of this: 1 over x plus 2, brackets x minus 5, plus 1 over x minus 3 times x minus 4 equals 0 as well; now, if we solve this one, for this one We can see that 2x is equal to 12, therefore x is equal to 6. Right then, for here we can say that 2x is equal to 7, therefore x is equal to 7 over 2. All right, now for this we are going to Solve this; this is going to take us a bit of our time, so let's solve it now. I'm going to transfer this to the other side of the equation, so if I do that, I will have 1 over; I will have 1 over (x + 2)(x - 5) okay is equal to remember that I'm transferring this On this side, I don't want to do it separately, wasting my time, so if I transfer, I get a minus of this so minus 1 over x minus 3, brackets x minus 4, okay. All right, now let us cross multiply if we Cross-multiplying, we have this; we carry the minus sign, so this side will be minus, brackets x plus 2. brackets x minus 5 is equal to x minus 3 brackets x minus 4 okay now let's open the bracket so if you open our brackets you have minus into x times x is x squared x times minus 5 is minus 5x plus 2 times x is plus 2x; plus 2 times minus 5 is minus 10. Okay, is equal to. Then here is x times x. x squared is x times x, minus 4 is minus 4x, minus 3 times x is minus 3x, and minus 3 times minus 4 Is plus 12 okay? All right. Now, permit me to draw my line. Okay. Oh, don't worry, I'll get a ruler soon. Alright, so I'm going to write down what we have here, which is the current situation or the present The level at which we are not in a situation, okay? Now this is where we are: that's minus brackets x squared. -5x + 2x - 10 is equal to x² - 4x - 3x plus 12, so this is what we have now. Now we have minus parentheses x squared minus 5x plus 2x we have minus 3x right then minus 10 equals x squared minus 7x plus 12 now let's use this minus and multiply out so we have minus x squared plus 3x plus 10 equals x squared -7x + 12. Now we are going to move this to this side. Okay, so if we do that, we have - x squared plus 3x plus 10 minus x squared plus 7x minus 12 equals zero. Now, if we solve, we minus x squared minus x squared gives us minus 2x squared; 3x plus 7x gives us plus 10x. 10 minus 12 gives us minus 2. Oh, oh, oh, excuse me. Now, immediately I got to that point, and this There must be a mistake somewhere, because I've solved this before; this is the mistake. Please let's correct it together. Now we have x minus 2 here, but unfortunately, I mistakenly wrote x plus 2 here, so we are going to change wherever we see this plus to minus 2. So this is x minus 2, so coming back to this side again, sorry, I've gone too far; I don't need to remove this. So this is minus 2, okay? This is minus 2, so that means this is minus 2 as well. Okay, now let's go. Back to where we are, so this is supposed to be minus 2 and not plus 2. Let's correct it again. Now that means minus 5x minus 2x will give us minus 7x, okay, minus 7x. So that means what we There is plus 7x here, plus 7x, okay, so we have plus 7x here as well. Now plus 7x plus 7x. will now give us plus 14x, all right? Okay, so um, this is minus 10, and this continues to be Alright then, also because we change this to minus, so minus times minus will give us Plus, so our 10 should be positive 10. Um, like I said, I've gone far; I can't afford to start over. Once again, please just bear with me—anywhere we see 10, it's going to be positive 10. So this is positive 10, this is positive 10, so minus times minus will give us positive. 10, instead, all right, so that means we have negative 10 here as well, negative, okay, so When we solve it now, we can see that minus 10 minus 12 will give us minus 22, instead of here, all right. and this is equal to zero; now let's divide through by minus 2 in order to remove this minus sign so if I divide everything by minus 2, I'm going to have x squared minus 7x plus 11 is equal To zero, right? I want to make sure it's correct. I don't want to make another mistake. All right. Now we are going to make use of the quadratic equation formula, which states that x is equal to minus b plus or minus the square root of b squared minus 4ac, all divided by 2a, okay, all right So now we have that x is equal to four; here, our a is equal to one, which is the coefficient of x. squared, which is one; our b is the coefficient of x, which is minus 7; while our c is equal to 11. Okay, so we can now substitute into the equation, and we have that x is equal to minus bracket. minus 7 right plus or minus the square root of now b squared, that is minus 7 squared minus 4ac, 4 multiplied by 1 multiplied by 11, okay, everything divided by 2 multiplied by So we have that x is equal to 7 plus or minus the square root of 49 minus 44, all divided by 2. Alright, therefore we also have that x is equal to 7 plus or minus the square root of 5. divided by 2, so that simply means another value of x is what, and this should be the third and the fourth value of x is equal to 7 plus or minus the square root of 5, divided by 2, and that is Hey guys, thank you so much for watching. Please subscribe to the channel if this is your first time. when you come across our page, our channel, and then on your notification bell so that you Don't miss our amazing videos, and don't forget to give us a thumbs up if you actually enjoyed this. video bye