Math Olympiad| can you find all values of k|Algebra

hi everyone let's solve for the six values of k so i would like you to watch to the end and enjoy the ride okay all right now the first thing we need to do is to bring this to this side yes so we have k to the power of six minus three to the power of six is equal to zero yes i mean most of you will be like k is equal to three yes of course k is equal to three from observation now what of the other five values of k you see why you need to watch the end all right now note that when you have a to the power of m multiplied by n okay this is equal to a to the power of m raised to the power of n all right which is equal to a to the power of m times n now having noted this we can express this as three times two so we write k to the power of three multiplied by two minus three to the power of three multiplied by two is equal to zero so they are reading this form now let me put them in this one because i want to form difference of two squares yeah so i can now write k to the power of three squared minus three to the power of three squared is equal to zero now we have difference of two squares which states that a to the power of two minus b to the power of two is equal to a minus b multiplied by a plus b okay all right now that simply means that we can express this as k to the power of three minus three to the power of three multiplied by k to the power of three plus three to the power of three is equal to zero okay now when you have two brackets multiplying each other and they're equal to zero simply means that either of them are equal to zero is equal to zero sorry so you have k to the power of three minus three to the power of three is equal to zero or k to the power of three plus three to 4 of 3 is equal to 0. Now here we are going to recall our algebra identity that states that a to the 4 of 3 minus b to the 4 of 3 is equal to a minus b multiplied by a squared plus ab plus b squared okay so that simply means that we can rewrite this as k minus 3 multiplied by k squared plus 3k 3 multiplied by k so let me not skip anything plus b squared that is 3 squared is equal to 0 so we can now write this as k minus 3 multiplied by k squared plus 3k plus 9 is equal to 0 so we now have two brackets again so we can say k minus 3 is equal to 0 or k squared plus 3k plus 9 is equal to 0 now from here we can see the first value of k which is 3 which is the one you can see by observation all right excuse me now when you come to this one we are going to make use of the quadratic equation formula to bring out the two values of k from here all right now that formula states that from here first of all we have a to be 1 that is the coefficient of k squared b to be 3 coefficient of k and c to be 9 are constant okay now the quadratic equation formula states that um this k we are looking for should be equal to a sorry it's going to be equal to minus b plus or minus square root of b squared minus 4ac everything divided by 2a all right so recall that our a is 1 our b is 3 and our c is 9 so let's substitute into this so we have that k is equal to um minus 3 plus or minus the square root of 3 squared minus 4 multiplied by a which is 1 multiplied by c which is 9 right everything divided by 2 multiplied by 1 now we have that k is equal to minus 3 plus or minus square root of 9 minus 36 right everything divided by 2 so we have k is equal to minus 3 plus or minus square root of minus 27 divided by 2 okay so we can see the second and the third value of our k here right so the second value of our k is so our k is equal to minus 3 plus or minus the square root of 27 can be expressed as minus 1 multiplied by 9 multiplied by 3 okay divided by 2 so we have that k is equal to minus 3 plus or minus square root of 9 multiplied by square root of 3 multiplied by square root of minus 1 divided by 2 so we have k is equal to minus 3 plus or minus square root of 9 will give us 3 and we have root 3 or right minus 1 that is i square root of minus 1 is i divided by 2 therefore the second and the third value of k is equal to minus 3 plus or minus 3 root 3 i divided by 2 so we have gotten the second and the third value of k now let us go back in order to get the fourth and the fifth and the sixth value of k now all those while we've been solving this particular question this one this equation now let us go ahead and solve this okay k to the power of 3 plus 3 to the power of 3 is equal to 0 so permit me to use this sign all right so we have k to the power of 3 plus 3 to the power of 3 is equal to 0 all right now we are going to use our um excuse me the algebra identity that states that a to the power of 3 plus b to the power of 3 is equal to a plus b multiplied by a squared minus a b plus b squared okay all right so this is going to be k plus 3 multiplied by k squared minus 3 multiplied by k plus 3 squared and this is equal to 0 so we have k plus 3 multiplied by k squared minus 3 k plus 9 is equal to 0 now to solve this we are going to use the quadratic equation formula but before that from here we can see that k plus 3 is equal to 0 and that k squared minus 3 k plus 9 is also equal to 0 okay so we can see that k is equal to minus 3 and this should be the fourth value of k right and also from here we can use we can see that a is 1 b is minus 3 c is 9 okay so we are going to use the quadratic equation formula that states that k should give us minus b plus or minus square root of b squared minus 4 everything divided by 2a so k is going to be minus minus 3 plus or minus square root of minus 3 squared okay um minus 4ac 4 multiplied by 1 multiplied by 9 everything divided by 2 multiplied by 1 all right have you subscribed to our channel please do that if this is the first tuning in turn on your notification bell so that you don't miss our amazing videos you upload videos at least twice daily so please tune in at your convenience to watch our videos all right and please give us a thumbs up if you're actually enjoying the video thank you so much so from here we can see that k is minus minus is plus so we have plus 3 plus or minus square root of minus 3 squared is 9 minus 4 times 9 is 36 divided by 2 so k is 3 plus or minus square root of minus 27 divided by 2 okay so we can see that k is equal to 3 plus or minus the square root of minus 1 multiplied by 9 multiplied by 3 divided by 2 so k is equal to 3 plus or minus square root of 9 will give us 3 root 3 then square root of minus 1 is i divided by 2 so right here you can see the fifth and the sixth value of k to be 3 plus or minus 3 root 3 i divided by 2 thank you so much for watching and bye