Harvard University interview question| You should know this trick|Algebra|Entrance question.

Hi! Welcome to my channel. Today, let's find the value of X in this particular question. Okay, first of all, regarding the fact that X can be factorized further, gives us insight into how to proceed with this. But I will ask, and I will tell you to proceed with caution. Yes, because this is a very tricky question. If not, you won't be able to determine the values of your X. Trust me. Now, let's get straight to the point. So, recall that A to the power of M, raised to the power of N is equal to A to the power of M times N. Okay? So, that means we can factorize X to the power of 6, x cubed squared Since 3 to the power of 2, Since 3 times 2 still gives us 6, right? Minus X cubed equals 2. Please, please, I've forgotten. Don't mind my writing. I am not left-handed. I'm supposed to write like this so I can feel comfortable. But since I'm right-handed, I'm forced to write like this. And I have to keep this up for some time. because I need to use more of my board. Okay? So, no matter how the writing is coming, Just take it like that. So, let's continue. Now, we have X to the power of 3, x cubed Let's call it something to simplify our equation. Okay? So, we say X to the power of 3, Okay. be equal to A Let's just call it A. Or let me use another alphabet because A looks like 9. Okay. So, it equals B. Alright. So, that means we can write this as what? B squared, right? Minus B to the... Minus B, sorry. B squared. Since X cubed is B, So it's B squared minus B equals 2. Now, that means we can write this as B squared minus B minus 2 equals 0. Now, let's go ahead and factorize this. So, we need two factors of minus 2. that when we add them, we get minus 1. That is the coefficient of B. And those two values should be 1 and minus 2. because 1 times minus 2 will give us minus 2 And 1 minus 2 gives us minus 1. So, that means we can replace minus B with plus B and minus 2B, right? So, let's do that. So, we have B squared plus B minus 2B minus 2 is equal to 0. Okay. Now, we can proceed to factor by grouping. Okay. So, if we group this and this, we have B. They have B in common. B squared divided by B gives us B. And B divided by B will give us 1, right? Minus, what do they have in common? They have 2. Now, 2B divided by... Sorry, minus 2B divided by minus 2 What will they give us? Plus B. And minus 2 divided by minus 2 gives us plus 1. It is equal to 0. Now, we have the same thing inside the parentheses. Okay. So, that means we can write this as what? B plus 1, alright, multiplied by B minus 2 is equal to 0. Okay. Now, it's time for us to enter the interesting part of this video. So, if this is the first time you're seeing this face, Why haven't you subscribed yet? Please subscribe to the channel, Turn on your notification bell, so that you'll be notified whenever I upload a new video. So, you don't miss my amazing videos. Okay. Alright. Now, this is what we have. Therefore, we can now say what? (B + 1)(B - 2) is equal to 0. So, we have two values of B here, right? So, we can say that B plus 1 is equal to 0. and that B minus 2 equals 0. Okay. I think it's time for me to turn. Now I'm comfortable. Okay. So, from equation 1, We can see that B is equal to minus 1. And from equation 2, We can see that B is equal to 2. As I said, proceed with caution. because we're not actually looking for the value of B. Recall that we are looking for the value of X. So, let's go ahead and get the value of our X. Now, the first value states that B is equal to minus 1. But recall that B, according to here B is equal to X cubed. So, B is equal to X cubed. Let me move so that you can see it very clearly. X cubed is equal to B. Okay. So that means that once we said that, Let's go for the first equation. If B is equal to X to the power of 3, for the first equation, we can say that X cubed is equal to minus 1. So, let's write that. X cubed is equal to minus 1. Now, how do you solve this? How? Okay. This is how we go about it because this is actually a polynomial. Now, this minus we carry over to become plus 1 is equal to 0. Now, when you look at this, You notice that we can express 1 as 1 to the power of 3. without changing its value, right? So, we can write X to the power of 3 1 to the power of 3 is equal to 0. Now, we are going to go back to our algebraic identity. Yes, the one that says that A cubed plus B to the power of 3 is equal to A plus B. Okay. Multiplied by A squared minus AB plus B squared. Okay. So, we are going to try to express this in this form. Okay. So, that means we can write What is this part of the equation? X plus 1, right? Because our A is X and our B is 1. So, we can write this as what? X plus 1 multiplied by Now, A squared That is X squared. So, we have X squared minus AB that is X times 1 which will still give us X plus B squared that is 1 squared which will still give us 1. Everything is equal to 0. Okay. Now, follow me to clean this because We don't really need it anymore. So, I'm going to draw my line right now And we continue. So, from here we have formed two brackets again. So, the first bracket is that X plus 1 equals 0. Okay. And what is the second bracket? x squared minus X Plus 1 is equal to 0. Now, from this side We can say that X is equal to minus 1, right? Okay. So, we can actually call this The first value of our X. Now, looking at this now, We cannot solve this by factorization, can we? So, let's go ahead and use the quadratic formula So, using the quadratic formula We can say that X is equal to minus B plus or minus This place is small. Okay. Now, we have that X is equal to minus B plus or minus square root of what? B squared minus 4AC everything divided by 2A So, we go ahead and solve that. X is equal to Our B is going to be the coefficient of X. Our A is the coefficient of X squared. B is the coefficient of X. YC is our constant. Okay. So, let's write that down. That A is equal to the coefficient of X squared which is 1. Our B is minus 1 coefficient of X And our C is 1 Which is the constant? So, let's substitute these values into this particular formula. So, we can now write that X is equal to minus B That is minus minus 1 That will give us plus 1, right? Plus or minus the square root of B squared That is minus 1 squared. which will give us 1. Then minus 4AC. That is 4 multiplied by A multiplied by C which will still give us 4. Okay. Divided by 2A. Recall that our A is 1. So 2 times 1 will still give us 2. Therefore, our X is 1. plus or minus the square root of what? -3 divided by 2 So, we can write that X is equal to 1 plus or minus Now, this is going to be root 3I. because we can express this as譬 √3 multiplied by √(-1) and the square root of 3 is the square root of 3. Why does the square root of minus 1 give us i? Divided by 2. Now, look at this. We have two values of X here 改编自原作的电视剧《大江大河》的第二部，讲述了改革开放初期，以宋运辉为代表的中国第一批改革者在经济、社会和思想层面的探索与奋斗。该剧通过细腻的叙事和真实的人物刻画，展现了那个时代背景下个体命运与国家发展的紧密联系。 the second value and the third value But for the fact that I am here This is not going to give us Real root, right? posal because of that We are going to leave this. and focus on this. We've got this, but remember I told you to be careful. Yes, we have obtained the first value. of the actual value of our X which is minus 1 but don't forget that All this time, have we been solving for what? B is equal to minus 1. That is what we have been working on. Now, we are going to solve for B is equal to 2. You see why I said you should Proceed with caution. So that we are able to get all the values of X I told you, didn't I? Now, let's quickly solve for what? B equals 2. Now, I'm going to clean from here. I'm only going to leave this or write it down somewhere that the first actual value is that X is equal to minus 1. Then, let me clean from here and then let's solve for 2 and check if we have real values there as well Okay, now this is what we have for B equal to 2. Recall that our B is equal to x cubed So, we can actually write that X cubed is equal to 2. Now, look at this. This is actually easy to obtain. And wait, we are still going to Substitute these values into our equation. to check whether what we have is correct. Okay, now look at it. We have X to the power of 3. Let's raise this to the power of 1 over 3 and this side to the same thing power of 1 over 3 Since we are looking for the value of X so that 3 we divide 3 And we have that. This is the 4th value, right? That the 4th value of X which is another real value 2 to the power of 1 over 3 Now, it's time to check. If you watched this far I just want to say A very big thank you for watching up to this stage and I would like you to write for me In the comment section, I did and that will make me know that Yes, this particular person Watched to the end. Thank you so much. Now, this is it. So, let's check our solutions The equation says X to the power of 6 minus X cubed is equal to to 2 and the 4th value Is that X equal to minus 1? So, we are going to have (-1)^6 minus minus 1 to the power of 3 is equal to 2. Now, when you when your negative number is being raised to an even number, when the power is Even so, you're going to get even. as your solution, right? So, minus 1 to the power of 6 will give us 1 minus, now this is odd number, 3. That means Our solution will still be negative. 1. So, we are still going to have minus 1 is equal to 2. So, 1 minus times minus will give us plus. So, we are going to One plus one is equal to two. and 2 is equal to 2. So, this is Actually correct. Now, that is the first one. Now, the main one is 2 to the power of 1 over 3, right? Let's see if that is correct as well. So, we got X to be equal to 2 to the power of 1 over 3 raised to the sixth power -2 to the power of 1 over 3 raised to the power of 3 is equal to 2. Now, 3 is going to divide What is this supposed to give us? 2 to the power of 2 minus this will divide this to give us only 2 is equal to 2. So, we have 2 to the power of 2 which is 4 minus 2, right? Is equal 2 and 2 is equal 2. Ta-da! There you have it. Thank you so much. for watching and I wish Have a lovely day.