Math Olympiad || Harvard University Admission Interview Tricks ||

She is given T square minus T cube is equal to 80. So here we have to find out the value of T. How we can find out the value of T? Let's see solution. T square minus T cube is equal to 80. This is question, right? So here we can write this T square minus T cube minus 80 is equal to 0. T square minus T cube minus 80. So this 80 we have to convert in two numbers that is in a square form and cube form the same number like T square minus T cube minus 4 4's are 16. So 4 square is 16. So we can write here minus 16. You can take in bracket. So here 16 and here plus 64. So 64 that is 4 cube 64 and 64 you can see and check is 6 plus 4 10 and here 6 plus 1 7 plus 1 8 minus we are taking outside. So minus 16 minus 64 that is total will be minus 80. 8 is equal to 0. Why we are taking this one because we have to take here a square form and cube form of 4. So that's why we are taking this way. Now T square minus we can write this one. T square minus T cube minus 16 minus 64 equal to 0. Now T square minus 16. And here minus T cube. Plus 64 equal to 0. So here minus T cube will become minus T cube minus plus minus so minus 64 equal to 0. Now T square minus this is 16. This is this section. We can write in 4 square form. So T square minus 4 square minus T cube plus 4 cube equal to 0 right now here T square minus 4 square and here cube and here 4 cube. So you can see this is a square form and this is cube form. So this is the formula of a square minus b square is equal to we know that a plus b times a minus b and this is the cube form. So a cube plus b cube is equal to a plus b times a square minus a b plus b square right now. We have to apply this two formula here. So we will get here first a square T square minus 4 square. So T plus 4 T minus 4. So it will become here T plus 4 times T minus 4 right now here next one is minus T cube plus 4 cube. So here a plus b a square minus a b so we can write this one. T plus 4 times T square minus 4 T. That is a b plus b square that is 4 square. Equal to 0 right now. We can write this one T plus 4. We are taking here common T plus 4 T plus 4. So we are taking common here. Now next term that is T minus 4 minus left term that is minus T square minus 4 T plus 4 square is 16. Now T plus 4 times T minus 4. minus T square minus minus plus 4 T plus 4 T minus 16 equal to 0 now. Now here T plus 4 times T minus 4. We can write this one T minus 4 minus T square minus plus 4 T minus 16 equal to 0 now. We can add this one T plus 4 times. Here T and here plus 4 T. So plus T plus 4 T will become here 5 T. minus T square minus 4 minus 16 minus 20. equal to 0. So here we got one equation that is 5 T minus T square minus 20 equal to 0. Now you can see this is T plus 4. So we can say here. We got here two equation first one is T plus 4 equal to 0 and next one is 5 T minus T square minus 20 equal to 0. So first equation is T plus 4 equal to 0 or next equation is 5 T minus T square minus 20 equal to 0. So from here from first equation T plus 4 equal to 0. Therefore, we can say value of T is equal to minus 4. So first value of T we got here that is T equal to minus 4 or you can say T1 is equal to minus 4. Now we have to solve this one 5 T minus T minus T square minus 20 equal to 0. So let's solve this one. Now we can write this equation as minus T square plus 5 T minus 20 equal to 0. If you are here multiplying with minus 1 so we will get here T square because minus minus will become plus and here it will become minus. So minus 5 T it will become plus because minus minus plus so plus 20 equal to 0. Now T square minus 5 T plus 20 equal to 0. We got one equation here. So your coefficient of T square that is A equal to 1. B equal to minus 5 and C is equal to 20. Now we have to solve this one. So we know the formula T is equal to minus B plus minus square root of B square minus 4AC upon 2A is equal to minus B value of B is minus 5. So minus 5 plus minus square root of B square is minus 5. minus 4 times A, A is 1 times C is 20 upon 2 times A 2 times A is 1 now is equal to minus minus plus 5 plus minus square root of 5 square 25 minus 20 portion 80 80 upon 2 is equal to 5 plus minus 25 minus 80. So here 10 minus 5 is 5 and here it will become 7 minus 2 is 5 and minus sign. So minus 55 upon 2 is equal to 5 plus minus we can write this one square root of 55 times square root of minus 1 divided by 2 now is equal to 5 plus minus square root of 55 times square root of minus 1 that is I square value of minus 1 is I square divided by 2 is equal to 5 plus square root of 55 and here square square root of I square. So square is cancelled from square root 55 I plus minus divided by 2 so we can say here. We got the value of T that is T2 and T3 two value of T first one is T2 is equal to 5 plus square root of 55 I divided by 2 and T3 T3 is equal to 5 minus square root of 55 I divided by 2 so we got here second value that is T3. So one is positive and one is negative but this is the imaginary solution. So let's verify for real solution. That is T1 is equal to minus 4 is this correct or not? So let's verify that value. So let's verify question is T square minus T cube is equal to 80. This is question right now. We have to substitute this value T1 is equal to minus 4. We have to substitute this value. So minus 4 square minus T cube T cube is minus 4 again. So minus 4 cube question mark 80 now 4 square so we can write minus 4 times minus 4 and here minus minus 4 times minus 4 times minus 4 question mark 80 now minus minus plus 4 for just 16 16 minus now here 4 for just 16 16 for just we can write a 64 so it will become minus minus plus and again minus so minus 64 question mark 80 now 16 here minus minus plus 16 plus 64 question mark 80 now 6 plus 4 6 plus 4 is 10 and here 6 plus 1 7 plus 1 8 80 is equal to 80 so here LHS is equal to RHS. So we can say value of T is equal to minus 4. This is correct solution and another we got another two value that is that that are the imaginary solution. So this is the solution. Thank you.