Olympiad Mathematics | Indian | Can You Solve This One?

Added: 2026-06-04

[00:00:00]Hi, everyone.

[00:00:03]Can you solve this one?

[00:00:07]Okay, so we have m squared minus m cubed equals 3 over 64.

[00:00:18]Okay, so how do we solve this?

[00:00:20]We are going to focus more on the right-hand side, okay?

[00:00:25]So we have m squared minus m cubed equals this 3 over 64.

[00:00:34]You know, we can write it as 4 minus 1 over 64.

[00:00:43]Okay. Now we proceed to go and write it as 4 over 64 minus 1 over 64. Still the same thing.

[00:00:59]Look at that. Still the same thing. And then we now have m squared minus m cubed being equal to m 4 over 64 is the same as 1 over 16.

[00:01:14]Then we have 1 over 64.

[00:01:19]So if you look at the left-hand side and the right-hand side, we can now express them in the same form.

[00:01:26]Yes, we can express what we have there in the same form.

[00:01:31]So that here we have m squared minus m cubed and is equal to here now. You know, we can write this as 1 over 4 to the power of 2.

[00:01:45]This is the same thing as 1 over 16.

[00:01:47]Then 1 over 64 is the same thing as 1 over 4 to the power of 3.

[00:01:56]Now the left-hand side and the right hand side can be m balanced.

[00:02:01]But before then, let's bring what we have on the right to the left so that we can have m cubed.

[00:02:09]Okay, we have m cubed.

[00:02:12]Oh, [snorts] this is m squared then minus m cubed.

[00:02:18]Then I'm bringing this here now, so this becomes minus 1 over 4 to the power of 2.

[00:02:27]Then this becomes plus 1 over 4 to the power of 3.

[00:02:33]And all of this will be equal to zero.

[00:02:37]Everything will be equal to zero. So, the next thing we'll do is to reposition. We're going to write m squared minus this is 1 over 4 in bracket, this will be squared.

[00:02:50]Then we have minus there we have m cubed.

[00:02:57]Then this is plus 1 over 4 again to the power of 3 being equal to zero.

[00:03:04]You have to pay attention to this um to what we are doing right now.

[00:03:10]Now, look at this. We can group what we have so that m squared minus 1 over 4 squared will be a group.

[00:03:21]This is a group now, right? Then we have minus we open this for this to be a group as well. So, we write m cubed and negative times positive is negative. So, here we have 1 over 4 to the power of 3 as a group and it's all equal to zero.

[00:03:42]Okay? So, look at that.

[00:03:45]So, at this point now, here we have difference of two squares and here we have difference of two cubes.

[00:03:54]Okay, so here we are. We are going to work this in two ways.

[00:03:59]I mean, separately. We are going to deal with this one first and then deal with this one after.

[00:04:04]For this one here, this is going to be giving us m from difference of two squares. We have m minus 1 over 4.

[00:04:16]into m plus 1 over 4 for the difference of two squares. So, this is what we have here.

[00:04:24]Then here, we are going to use um difference of two cubes, right? And we know that a cubed minus b cubed can be written as a minus b into a squared plus ab plus b squared.

[00:04:46]Right?

[00:04:47]And according to this now, our a is m.

[00:04:51]Okay? Our a is m.

[00:04:54]And I'm writing this here.

[00:04:56]a minus b is now m minus 1 over 4.

[00:05:02]Then into a squared is m squared.

[00:05:06]plus a a is m.

[00:05:10]Multiply by b. b is 1 over 4. I'm working from the difference of two cubes there.

[00:05:17]So, b is 1 over 4. Then plus here, we have b squared. And that is going to be 1 over 4 squared, which is 1 over 16.

[00:05:28]So, we do this.

[00:05:31]Now, we are going to bring this and this together, right? And we are taking the difference between the two of them.

[00:05:41]So, let's set it very well.

[00:05:46]Okay, so now we are going to have um um m - 1 over 4 into m + 1 over 4. This is for the difference of two squares. My does for the difference of two cubes. Now we are writing this one. Everything here.

[00:06:06]So we open bracket, we write m - 1 over 4.

[00:06:10]Right? And this is going to multiply this, which is m squared + Okay, m * 1 over 4, that's the same as m over 4.

[00:06:23]Then plus um 1 over 6, right? 1 over 16.

[00:06:30]So this is what we have. And we equate everything here to zero.

[00:06:36]Everything will be equated to zero. Now looking at this, we have a common factor, which is m minus 1 over 4. So let m - 1 over 4 come out as a common factor.

[00:06:52]Then here I have m + 1 over 4.

[00:06:58]Then this negative will open the whole of this. Remember this is out already.

[00:07:03]So this negative will open this and we'll have negative m squared.

[00:07:09]Then we have negative m over 4 and negative 1 over 16.

[00:07:16]So this is equal to zero.

[00:07:20]Interesting.

[00:07:21]So we continue from here.

[00:07:25]What do we do at this point?

[00:07:29]Okay, before we simplify this, before we simplify everything here, let's apply our zero product rule. So we are saying that m - 1 over 4 is equal to zero or this one here is equal to 0.

[00:07:45]But, let's get our M from here first.

[00:07:48]So, M from here will now be 1/4 if we collect like terms.

[00:07:54]So, we now have the value of M.

[00:07:57]Okay? Now, to get the other solutions, we will bring down this um expression here.

[00:08:06]Let's bring it down here.

[00:08:07]We have M + 1/4 - M squared M over 4 - 1/16 and we will equate to 0.

[00:08:24]Okay, so let me pick this and then solve this completely.

[00:08:30]Okay, so from here now we will simplify what we have. The LCM is 16. 16 * M, that will be 16 M.

[00:08:40]Right? Then, we have plus 16 / 4, that is 4. 4 * 1 is 4.

[00:08:47]Then, - 16 * this is 16 M squared.

[00:08:53]Then, 16 / 4 is 4.

[00:08:57]4 * M, that will be - 4 M.

[00:09:02]Then, here we have - 1 cuz 16 / 16 is 1.

[00:09:07]1 * 1 is 1. And everything is equal to 0.

[00:09:11]So, let's rearrange what we have here.

[00:09:14]16 M - M, that will be 12 M.

[00:09:19]Okay?

[00:09:20]Then, we have 4 - 1, that will be + 3.

[00:09:25]Then, we have - 16 M squared which is equal to 0.

[00:09:31]Now, we are going to write this one first. - 16 M squared + 12 M plus three equals zero.

[00:09:43]So, we now have a quadratic equation.

[00:09:45]Multiply all through by -1.

[00:09:48]So, that will give 16 m squared to make this -12 m, make this -3, and everything is equal to zero.

[00:09:59]So, from here, what should we do?

[00:10:01]We will apply um we'll use quadratic formula for this.

[00:10:05]So, our A is 16. That is the coefficient of m squared.

[00:10:11]Then, [snorts] our B is minus 12.

[00:10:16]Coefficient of m. And C is the constant, which is minus three.

[00:10:22]C is a constant, which is minus uh three.

[00:10:27]Okay.

[00:10:29]So, we are using the quadratic um um formula, which is m equals minus B plus or minus we have B squared minus 4 AC.

[00:10:42]And this is over 2 * A.

[00:10:45]So, once you know the formula, you put in your values of ABC.

[00:10:50]So, our m will be negative B is supposed to be 12, right? Minus 12.

[00:10:56]So, negative negative will give us just positive 12.

[00:11:01]Then, we have plus or minus we have here Okay, let's jump a step, right? Here, we have B squared, which would be minus 12 squared, which would give us 144.

[00:11:12]4 Okay.

[00:11:14]To give us 144, then we have minus Well, four our four Okay, let me write that.

[00:11:23]We have 4 * A, which is 16, then times C. C is minus three.

[00:11:31]So, this is all over two multiplied by 16 because A is 16.

[00:11:39]And now from here, our M is equal to we have um 12 plus or minus 144 plus cuz negative negative is positive and 4 * 16 * 3 will give us 192. So, we have 192.

[00:12:04]And all of this is over two.

[00:12:08]So, our M will now be 12 plus or minus square root of 336.

[00:12:16]This is over Okay, this is 32.

[00:12:20]Yes, because 2 * 16 is 32. So, here we have everything over 32.

[00:12:28]Now, let's break our 336.

[00:12:32]Okay, so to break that, we're going to have M being equal to we have 12 plus or minus square root of 16 * 21.

[00:12:46]That will give us the same 336. This is divided by 32.

[00:12:51]Okay, now let's go on from here.

[00:12:55]Okay, so um from here, M will be 12 >> [sighs] >> plus or minus square root of 16 is four, then multiply by 21.

[00:13:07]Remember, everything here is over 32, right?

[00:13:13]Now, there's something else we can do.

[00:13:16]Our M will be four into three plus or minus this four is already out. We have square root of 21.

[00:13:26]And everything is over 32.

[00:13:31]Okay, so that from here M will be 4 into 32 is 8. So, we have 3 plus or minus square root of 21 over 8.

[00:13:43]So, this right now is our value of M.

[00:13:47]But, this is a two-in-one value.

[00:13:51]Right? Now, the three solutions that we have now are M to be equal to 1/4 from the first part.

[00:14:00]Then, M again is from here 3 plus square root of 21 over 8.

[00:14:10]Right? Then, our third solution Our third solution will be 3 minus the square root of 21 over 8. So, these are the three solutions to the equation.

[00:14:26]Thank you for watching.