Olympiad Mathematics | Indian | Can You Solve This?

Ajouté : 2026-05-09

[00:00:01]Okay, you're welcome.

[00:00:03]If you're ready, let's provide the solution to this one.

[00:00:09]We have x -1 to the^ 3 to be equal to -8.

[00:00:18]Okay, this is um a simple one. The first step is to bring this to the left. So we have x - 1 to the^ 3 + 8 and it's now equal to zero because we have shifted - 8 to the left and it becomes positive.

[00:00:46]Now this 8 can be written to half power of 3. Right? So that means that we have x - 1 to the^ 3 + 2 ^ 3 to be equal to zero.

[00:01:03]And this is what we call the addition of two cubes.

[00:01:08]And this is an identity in mathematics as well. If you have a cube plus b cube, this can be expressed as a.

[00:01:23]Okay, we have a + b in brackets and it's multiplying a squ - a + b 2.

[00:01:40]Okay. So this is the identity we call addition of two cubes. And from here we know what to do already because our a is the whole of this x - one and our b is 2. So let's put them into this um identity. So we have x -1.

[00:02:03]Then b is going to be + 2. Okay. Okay.

[00:02:07]So this is what we have in the first bracket. Remember x -1 is a and b is 2.

[00:02:15]So we go over to that and we write a 2.

[00:02:19]a 2 is going to be x -1 squared. Right? Then minus a here is going to be open bracket x - one as well.

[00:02:32]And it's going to multiply b and our b is 2. So we put two here. Then we have + 2^ 2, right? Because b is 2 for this particular equation.

[00:02:49]And we equate everything to zero.

[00:02:55]Now we are going to continue from here.

[00:02:58]From here, we're going to have x + 1 because -1 + 2 is 1. Then we have the expansion of this, right? So the expansion of this is going to give us x², right? Then we have - 2x as we have + 1. So the expansion of this alone will give us this. Right? If I like I can still put this in brackets although I have expanded it to get this.

[00:03:38]Then here I'll have minus then two or this one will multiply this.

[00:03:46]So x * 2 it will still be in brackets.

[00:03:51]So x * 2 that will be 2 x. Then -1 * 2 that is -2 in bracket and we have + 4.

[00:04:04]Okay. So everything here is still equal to zero.

[00:04:11]Yeah. Let me set this very well. So from here now what we'll do is to simplify what we have here. So I'm going to pick x + 1.

[00:04:24]Then here we open the bracket as we have x² - 2x + 1. Then we have - 2x.

[00:04:36]This will turn to + 2. Then we have + 4.

[00:04:42]Remember everything is equal to zero.

[00:04:48]Okay? If you're getting it, you have to subscribe to my channel. Now, we have x + one here.

[00:04:57]Then here, we only have 1 x^2. So, write your x².

[00:05:03]Then here we have - 2x - 2x. That will turn to - 4x, right? Then we have 1 + 2 + 4. and that is seven. So we write + 7.

[00:05:21]So we equate to zero.

[00:05:25]So from here we have our zero product rule. So either of them either of the two factors will be equal to zero. So let me work with this first. As I say x + 1 is equal to zero. I will come back to pick this one.

[00:05:46]And this means that our x is = 0 -1.

[00:05:51]And at the end of the day, the value of x is -1.

[00:05:57]This is our first solution.

[00:06:00]And we are going to get two more solutions from here. This is a quadratic equation. So let me pick it out very quickly.

[00:06:13]Okay. So this is the other factor and we have equated it to zero and it's a quadratic equation.

[00:06:23]We're using quadratic formula for this which is - b plus or minus we have b^ 2 - 4 a c all over 2 a.

[00:06:36]So if we go on we have to get our ABC.

[00:06:41]A is a coefficient of X squ which is 1.

[00:06:45]B is a coefficient of X and that is -4 and our C is 7. So our X will now be now we're supposed to have -4 but that will turn to + 4. then plus or minus b^ 2 that's going to be -4 in bracket squar then -4 * 1 then * 7 yes we have 7 for c and this is all over 2 * 1 so to go on with this our x will 3 4 + or minus - 4^ 2 is 16 - 4 * 7 is 28.

[00:07:42]So we have this over 2 cuz 2 * 1 is 2.

[00:07:50]Okay. So from here our x is going to be 4 + or minus square<unk> of -12 and this is over 2. 16 - 28 is -12.

[00:08:07]But there's something we can do here.

[00:08:10]This is 4 + or minus<unk> 12 *<unk> -1 / 2. Okay, we have to split it this way so that we can simplify roo<unk>2.

[00:08:27]And from here we have our root our x to be equal to 4 plus or minus the square root of 12 is 4 * 3<unk> of -1 is i. This is all over 2.

[00:08:46]So if we proceed then we will have okay we are going to have x to be equal to 4 plus or minus square root of 4 is 2 * this i. So we have 2 i but this three is still under the root. So we write roo<unk>3 and everything here is over two.

[00:09:16]And um now from here we have x to be 2 into 4 that is 2 plus or minus 2 into 2 that will give just i and then we have <unk>3 but this is a two in one solution.

[00:09:35]But I want us to bring the three solutions together. So stay to the end.

[00:09:41]Remember we got x to be equal to minus1.

[00:09:47]So this will be our first solution.

[00:09:51]Then the second solution x2 is from here 2 + i <unk>3. This our second solution.

[00:10:01]Then the third x3 is from here 2 and we have 2 - i <unk>3. So these are the three solutions to this particular equation.

[00:10:16]Thank you for watching.