Olympiad Mathematics | Indian | Can You Solve This?

Added: 2026-05-06

[00:00:00]Okay, if you're ready, let's provide the complete solution to this problem very quickly.

[00:00:12]We have x to the^ 3 to be = 8 to the^ of -1.

[00:00:20]Now we are having three solutions from here. We're going to have three solutions because of the power of three.

[00:00:28]Now here is what we should do. This is x to the^ 3 = 8 ^ -1 is the same as 1 / 8.

[00:00:38]Okay. And then from here we are going to express 1 / 8 in the power of three.

[00:00:47]Okay. That's very possible. So write x ^ 3 to be equal to um from here we can write um 1 / 8 as 1 / 2 to the power of 3.

[00:01:06]Yes. Okay. So this is very very possible because 1 to the^ 3 is 1. 2 to the^ 3 is 8. Now bring this to the left hand side as you have x ^ 3 - 1 / 2 to the same power 3 equals z.

[00:01:28]So at this point you just apply your difference of two cubes.

[00:01:34]Yes. And it says that a cub - b cub is equal to a - b * a 2 + a b + b 2.

[00:01:54]Yes, this is what it is. So that our a is going to be x. Then our b is going to be 1 / 2. This will be in the first bracket. Then in the second bracket a 2 will turn to x².

[00:02:13]Then + a. A is still x right. Then multiply by b which is 1 / 2.

[00:02:20]Then + b 2 and that is going to be 1 / 2. You know this will be squared.

[00:02:29]Okay. Let me turn this.

[00:02:32]Okay. So, this is it.

[00:02:35]And um we're going to equate everything here to zero.

[00:02:40]Yes. Everything will be equated to zero.

[00:02:45]So, now we have our x - 1 / 2 one of the factors. Then from here we have x² and then x * 1 / 2. that will give us um x / 2 + 1 / 2^ 2 will give 1 / 4.

[00:03:05]Okay. So this is equal to zero.

[00:03:11]From here what do we do? Let's um okay I think it's okay for us. We can apply our zero product rule already so that either this is equal to z or this is equal to z. So let me pick this. I'll come back to this one after x - 1 / 2 = 0.

[00:03:35]And this means that x = 0 + 1 / 2 and that is 1 / 2. So we have our value of x already to be 1 / 2.

[00:03:49]But then I'm going to pick out this equation and then we'll solve it.

[00:03:56]Okay. So, here it is. And remember that this is still a quadratic equation. Yes, even if we having fractions, it is a quadratic equation. And we can clear out the fractions. How do you do that? The LCM of everything is four. So, multiply four by all the terms. 4 * x 2 that will give us 4 x^ 2 + 4 * x / 2 that will give 2 x. Yes. Then 4 * 1 / 4 will give 1 which is equal to 4 * 0 and is 0.

[00:04:34]So from here now we have 4 x².

[00:04:40]Okay. I think we already have what we want.

[00:04:44]Yes, this is already our quadratic equation and we are going to solve it.

[00:04:48]Let me remove this. We're going to solve this formula using our quadratic um we're going to solve this equation using our quadratic formula that says x = - b + or minus we have the square root of b^ 2 - 4 a c all over 2 * a.

[00:05:15]Now what is our a? Our a is 4. Our b is 2 and then c is 1. a is a coefficient of x².

[00:05:29]b is a coefficient of x and c is a constant. So we're going to put the three of them into the formula.

[00:05:38]So our x will now be -4 + or minus we have b ^ 2 which will be 4. Okay.

[00:05:48]What is our b? b is 2. So we write 2^ 2 there. Then - 4 * 4 * 4 cuz a is 4 then * 1 c is 1 and all of this is over 2 * 4 since a is 4.

[00:06:14]So our x will now be -4 plus or minus we have 4 - 16 4 - 16 and it's all over 8.

[00:06:30]From here we have x to be -4 plus or minus the square root of 4 - 16 is -2.

[00:06:40]So this is all over 8. Now let us um work on root -12.

[00:06:48]Let's work on that root. So that x will be -4 + or minus square<unk> of 12 *<unk> of -1.

[00:07:00]So that if you multiply both of them you have roo<unk> -12 all over 8. Now to go on with what we have our x will be -4 plus or minus square root of 4 * 12 is 4 * 3.

[00:07:21]Then square root of -1 is i and all of this is over 8. Okay. So from here now we can continue to get x = -4 + or minus square root of 4 is 2 2 * i that will be 2 i then we have <unk>3 as we divide all of this by 8.

[00:07:50]Okay. Now let's break it very quickly.

[00:07:56]Okay. So we break it down to get -4 / 8 + or minus we have 2 I <unk>3 / 8. So what are we saying that x is = -1 / 2 plus or minus um 1 / 4. By the way that should be I yes I over 4. Then we have <unk>3 and this is a two in one kind of solution. So to get the full solution together we have x to be 1 / 2 if you remember. So this is our first solution.

[00:08:42]Then the second x2 is - 1 / 2 + i / 4 <unk>3. This is the second. Then the third is -1 / 2 - i / 4 <unk>3.

[00:09:03]So these three are the solutions to the equation. Thank you for watching.