A math Olympiad algebra question|Algebra simplification| Find all values of x

Let's solve for k, guys. Let's move this constant to the other side of the equation, so when we do that k squared minus k cubed minus 2 divided by 27 equals 0 Now I want us to note that 2 over 27 is the same thing as 3 over 27 minus 1 over 27, right? so let's substitute this for this; now we are going to write k squared minus k 3 minus 3 divided by 27 minus 1 over what means 7 is equal to 0, okay? All right, so we have k squared minus k cubed minus 3, we divide it by itself to get 1, we divide 27 to give us 9, so we have minus 1 over 9 minus 1 over 27 equals 0. All right now. we have k squared; let's bring the squares together minus 1 over 9, then minus k squared 3 minus 1 over 27 equals 0. Yeah, I brought them together because I know this is going to give me squares and this just when this will give me cube, right? Okay, so we can now write, excuse me. So we can write k squared minus 1 is the same as 1 squared, and 9 is the same as 3 squared. Okay, then minus k cubed minus 1 cubed cubed is equal to 0. Now I want us to note that a to the power of x divided by b to the power of x is equal to a. divided by b, everything raised to the power of x. Okay, so that means that we can write this as what? That means we can write this as (k - k² - 1/3)², okay. minus k cubed minus 1 over 3, all raised to the power of 3, is equal to All right, now we have a difference of two squares here, and we have a difference of cubes here, okay. now there is this identity in algebra that says that a squared minus b squared is equal to a minus b multiplied by a plus b, okay, okay, and this one cube, that is a to the power of 3 minus b to the power of 3. just as we have here, a minus b multiplied by a squared plus a b plus b squared Alright, so we're going to express this and this in this form. Now, let's go ahead and do that. so that means we can write this as (k minus 1) divided by 3 multiplied by (k plus 1) divided by 3, okay? Now minus. Then, for this place, we have what, k? Okay, our first bracket is going to be k minus 1 over 3, all right? Now we are going to multiply this by what k, that is a squared. is k squared plus a b, which is k times 1 over 3, which will give us k over 3, right? Then plus b. squared that is 1 over 3 squared, which will give us 1 over 9. All right, now everything is equal to 0 so when you look at this side of the equation and this side of the equation you will notice that we have k minus 1 over 3 here and k minus 1 over 3 here as well, so let's factor that out so we have k minus 1 over 3, okay, multiplied by so when the whole of this is divided by this, this will remain right, so we have k plus 1 over 3 minus now when the whole of this is divided by this, this will go, and we are left with what k squared plus k over 3 plus 1 over 9 Alright, now everything is equal to zero. Now, if this is the first time you're coming across this channel Please click the subscription button and turn on your notification bell so you'll be notified. Whenever I upload a new video, thank you so much, and please kindly give us a thumbs up. if you're actually enjoying this video, okay, so we have that from here we can say that k minus 1 over 3 is equal to 0, right, and the entire expression is also equal to 0, but before that, let me try to simplify this bracket further so we have k minus 1 over 3, all right, multiplied by k plus 1 over 3 minus now minus times k squared will give us minus k squared minus times plus k Over 3 will give us minus k over 3, and minus times plus 1 over 9 will give us minus 1 over 9. Right now everything is equal to 0, so we have k minus 1 over 3 multiplied by. Now let's collect Like terms; if we do that, we have k minus what, k over 3? Now, k is nothing but k over 1, right? then we have a minus k squared, it doesn't have any term that relates to it or It doesn't have a common term, okay, so this is minus k squared. Now when you look at this, you See that we have a constant and a constant, so we have plus 1 over 3 minus 1 over 9, all right. Now everything is equal to 0, so we have k minus 1 over 3, which is nothing compared to k over 1. All right, multiplied by now, when this is for me to solve this, this has to have the same denominator. Yes, that's right. So I have to multiply this by 3. So 1 times 3 means I'll multiply the numerator. multiply by 3 again to give me 3k, right, minus k over 3 minus k squared, plus the same thing here to make This to be 9, I'll multiply it by 3, so 3 times 3 gives us 9. 1 times 3 will give us 3 minus 1 over 9 All right, now everything is equal to 0, so we have k minus 1 over 3 multiplied by 3k minus k. will give us 2k divided by 3 then minus k squared, all right? I would like to write k squared first. because this is what a quadratic equation is supposed to look like—the k squared term should come first so minus k squared; now 3k minus k will give us plus 2k divided by 3; now plus 3 minus 1 will give us plus 2 divided by 9 now everything is equal to 0 okay now if you look at this you see that we can now set this equal to 0 and set the other side of the equation equal to 0, so we can say that k minus 1 over 3 is equal to 0 or that minus k squared plus 2k over 3 plus 2 over 9 is also equal to 0 now; if we solve the left-hand side, on this side we are going to have that k. Is the first value of k equal to this we cross to become 1 over 3, so this is the first value of k. Now, for the second value of k, we have to remove this negative sign, and to remove it, we multiply Multiply through by minus 1, so what we do is, this side becomes k squared minus 2k over 3 minus 2 over 9. верно, сейчас равно 0, вот что я сейчас решу, хорошо, давайте решим это вместе Alright, so we have that k squared minus 2k over 3 minus 2 over 9 is equal to 0. Okay, now I want us to recall the quadratic equation formula, which states that In this case, the value of k we are looking for should be equal to minus b plus or minus square root of b squared minus 4ac, all divided by 2a. All right, now our a is the the coefficient of k squared, which is 1, our b is the coefficient of k, which is minus 2 over 3 and our c is our constant, which is minus 2 over 9. Permit me to clean everything—the only thing What I have here is that k is equal to this, so everything else is unnecessary. Alright, so what I see is our constant which is minus 2 over 9, okay. All right, so let me write this somewhere that the first The value of k is 1 over 3. All right, now we can substitute into this quadratic equation, so when We do that; we have that k is equal to minus minus, which will give us plus 2 over 3, right? plus or minus the square root of now b squared, which is minus 2 over 3 squared Okay, minus 4 multiplied by a is 1, okay, so no need to put it here, and our c is minus what? 2 over 9, all right. Now, everything divided by 2a, which is 2 times 1, which will still give us 2. Okay, so we can write that our k is equal to 2 over 3 plus or minus the square root of Now minus 2 squared gives us 4, and 3 squared gives us 9. Okay, minus now 4 times minus 2. we give ourselves minus 8, and this will give us minus 8 over 9; right now everything is divided by 2, so let Rewrite what we have here because I don't know if you can see everything, so that means our k is equal to 2 over 3 plus or minus the square root of I'm repeating what we have here 4 divided by 9 minus minus 8 over 9, okay? Now everything divided by 2. All right, so that means that our k is 2 over 3 plus or minus the square root of now we have 4 over 9 minus minus is close All right, everything divided by 2, okay, so our k then is equal to 2 over 3 plus or minus. The square root of 4 plus 8, since we now have the same denominator, 4 plus 8 gives us 12. Divided by 9, everything divided by 2, okay, so we now have that k is equal to 2 over 3. okay plus or minus now the square root of 12 is the same as the square root of 4 multiplied by The square root of 3 and the square root of 9 is 3, right? Divided by 2, all right, please. Allow me to clean this as well. Now that we have this, our k is equal to 2 over 3 plus or minus Now, the square root of 4 gives us 2, right? So we have 2 multiplied by root 3, which is still root 3. divided by 3 right now, everything divided by 2. All right, so what we have now is that our k is equal to 2 over 3 plus or minus oh no need to repeat that, no need to repeat that That means what we have here is that k is equal to 2 over 3 plus or minus 2 root 3. divided by 3, all right, everything is divided by 2 now I'm going to change this and then do the same for this as well, because this is the same as 2 over 1, so our k is what 2 over 3 plus or minus 2 root 3 over 3 times 1 over 2, okay? Now when you look at this side of the board When you look at this side of the board, you notice that we can actually factor out a 2. Right, yes, we can factor out 2, and when we do that, we have that k is equal to, let's bring it out. So what remains here is 1 over 3 plus or minus what remains here is root 3 over 3. right times 1 over 2; now these two can cancel, these two, right? yes, and when that happens, we we are left with the fact that our k is equal to 1 over 3, 1 over 3 plus or minus The square root of 3 divided by 3, which is also the same as 1 plus or minus the square root of 3. everything divided by 3, so these are the second and third values of our k, right? So the second the value is that k equals 1 plus root 3 over 3, while the third value of k is that k equals 1 minus root 3 over 3, thank you so much for watching