Find X! A tricky algebra and factorial challenge.

No, to solve for x looks impossible, but it is very possible. Welcome back to my channel. First thing we need to do here is, I want you to note that a to the power of m plus n is equal to a to the power of m multiplied by a to the power of n. So, let's rewrite this in another form. We have x factorial is equal to x to the power of, I can write it as 2 plus 1, right? Minus x. X factorial is equal to, then I can expand it to this one. X equal to, multiplied by X equal to 1, which is still X, minus X. Alright? So, we have X factorial is equal to, if you look at this, you see that both of them share X in common. So let's factorize X. This, we go into X, and X where do we remain? And X, we go into X, and 1, where do we remain? Please subscribe to this channel, turn on your notification bell, if this is the first time you have seen this lovely channel. Okay? We upload our videos as often as possible, 2 minutes of content, and most of it, something new for you to watch. We need your comments. Tell us what you feel, how you feel the channel is. The good, the bad, the ugly, we will appreciate that. Now, I want you to recall that. x factorial is the same thing as x multiplied by x minus 1 multiplied by x minus 2 multiplied by x minus 3 factorial. Now, I'll explain, okay? Remember that, for example, 4 factorial is the same thing as 4 multiplied by 3 multiplied by 2 multiplied by 1, okay? And 3 is the same thing as 4 minus 1, which is what we have here. 2 is the same thing as 4 minus 2, which is what we have here. 1 is the same thing as 4 minus 3, which is what we have here, okay? But let's assume I want to stop this at 3 only. Then I need to add my factorial. Yes, because 3 factorial still remains 3 times 2 times 1. So from here to the end, from here to the end is what 3 factorial. So anyway, I decide to stop. I have to put factorial in this for this to be correct, okay? Now, I'm going to explain that. Let's continue. We can now rewrite this as x multiplied by x minus 1 multiplied by x minus 2 brackets x minus 3 factorial, okay, is equal to x brackets x squared minus 1. Alright, now, recall that this can be written as a squared and a squared minus b squared is equal to a minus b multiplied by a plus b. So, that means that we can rewrite this as x brackets x minus 1 brackets x minus 2 brackets x minus 3 factorial is equal to x brackets. Now, we're going to express this as the difference of two squared, so it's going to be x minus 1 multiplied by x plus 1. Let's divide both sides. So if we do that here, this we go and we are left with this, x minus 2 multiplied by x minus 3 factorial is equal to, and for the right hand side, this and this we go as well, and we are left with x plus 1. Now, let's transfer this to this side. So now we have x minus 2 brackets x minus 3. factorial minus x is equal to 1. Now we are going to add 2 to both sides of the equation. I'm looking at this negative 2 because I want to factorize that out from this side. Okay? So if we do that, let me clean everything together. All right. Now if we do that, we are going to have x minus 2 bracket x minus 3 factorial Minus x plus 2, add 2 to both sides, so I'm adding 2 to both sides, so is equal to 1 plus 2 as well, alright? So, here we have x minus 2 into x minus 3 factorial minus, this becomes x minus 2 as well, because minus minus is plus, okay? Now, let's factorize x minus 2 out. So we have x minus 2, open bracket, then we have x minus 3 factorial minus 1, right? Okay, is equal to 3. It's telling us something, but wait first, we can write, but 3 is equal to 3 multiplied One by one, right? Therefore, we can rewrite this as x minus 2 multiplied by into x minus 3 factorial minus 1 is equal to 3 multiplied by 1. And if we say that, that means we have a new equation coming. Let's just clean this. Okay. Now, we can see that we can say that minus 2 is equal to 3, right? And we can say that this other bracket, that is x minus 3 factorial minus 1 is equal to 1. It will make sense, right? Okay, now, if we solve for that, we will have x is equal to minus 2, we cross, we come plus 2. So, we have 3 plus 2. And from here, we have x minus 3 factorial is equal to 1, minus 1, we cross, we come plus 1. Are you enjoying this? Because definitely, me personally, I am enjoying this. Okay, so we have x equals 5 here. And from here, we have x minus 3 factorial is equal to 2. So this is the first value of x we are seeing, right? And for this one, we can say x minus 3 factorial is equal to 2 factorial. Mm-hmm. Mm-hmm. Because two... is also equal to two times one which is two factorial. So if I say that two is two factorial I am 200% correct. I have not changed anything. Now by comparison, by comparison we can say that x minus 3 is equal to 2 right. So if this crosses we have x equals 2 plus 3 And we have x equals 5. Wait, wait. We still have other options to explore. Let's go back to this. If you look at this, we can say that x minus 2 is 1, right? And x minus 3 factorial minus 1 is 3. We can actually interchange the product. Let's see if that is the actual answer. So, if we said that, so we have gotten x to be 5 twice, right? Okay. Here we can now say x minus 2 is equal to 1 and x minus 3 factorial minus 1 is equal to 3. So if we solve the left side we have x equals 1 and this crosses to give us plus 2. So our x equals 3. Now let's check this side. Here we can say x minus 3 factorial is equal to 3. This we cross to become plus 1. So, we have x minus 3 factorial is equal to 4. Now, I want us to check something. Do we have any factorial that is equal to 4? Let's check. 2 factorial is 2. 3 factorial is 6. 4 factorial is 12. Right? Okay. No, 4 factorial is 4 times 3 times 2. 4 times 3 is 12. 12 times 2 is 24. Right? Now, 3 factorial is 3 times 2 which is 6. So, this is not possible. Impossible. It's not possible. And that disqualifies these two. Now, it simply means that x is what? Equal to 5. Now, let's prove x equals 5. Now, let's prove that. We have x factorial, which is 5 factorial, is equal to 5 to the power of 3 minus 5, right? That is if x equals 5. Now, 5 factorial is 5 times 4 times 3 times Times two, times one. And five to the power of three is one, two, five. So, we have 5 times 4 is 20, 3 times 2 is 6. Now, 20 times 6 is 120, right? So, we have 120 is equal to, 125 minus 5 is also 120. And this proves that the answer we got is correct. Please share this too. to students that you feel need this for any exam at all and thank you for your support to all my old subscribers thank you so much and see you in my next video bye