Olympiad Mathematics | Indian | Can You Solve This?

追加: 2026-05-11

[00:00:01]Hi, everyone.

[00:00:04]If you are ready, let me show you the two ways you can solve this problem here.

[00:00:11]Solution.

[00:00:15]We have the square root of 3 m over m equals 2.

[00:00:24]Okay, the first method we're going to apply is to cross multiply because we know that this is over 1.

[00:00:34]Yes, we know that this is over 1. So, we are going to have um square root of 3 m to be equal to m multiplied by 2.

[00:00:49]And then the square root of 3 m will be equal to 2 m.

[00:00:56]Okay, so what again do we do from here?

[00:01:01]Very quickly, we are going to square both sides of the equation.

[00:01:07]In case you do not know, we are squaring both sides so that the square root here will go.

[00:01:13]And if you do that to the right, you have 2 m squared.

[00:01:21]Okay. Now, you have to put this um under the square root sign.

[00:01:27]Sorry, you have to put this in bracket, right? If not, you will be making a mistake.

[00:01:33]For example, if you write this as 2 m squared, it means that the square is for only the m and it has nothing to do with the 2.

[00:01:45]Okay, so it will be wrong.

[00:01:47]Now, let me remove this so that we can continue.

[00:01:52]>> [snorts] >> Right? Now, from here now, this is taking this one out and 3 m is on the left-hand side, which is equal to the square of 2 m, which will give us 4 m squared.

[00:02:08]2 squared is 4, and then m squared is um coming down, right?

[00:02:16]Remember, this is This can be written like this.

[00:02:20]2 squared * m squared. Okay? So, this is what give us this.

[00:02:31]>> [snorts] >> Okay, so, we're going to continue from here as we write this one first before this one. So, 4 m squared is equal to 3 m.

[00:02:43]But, we'd like to bring them to the same side.

[00:02:46]And our 4 m squared - 3 m will be equal to 0.

[00:02:54]So, this is now a quadratic equation, and we're going to get two solutions already.

[00:03:01]Mind you, m is the only common factor.

[00:03:05]Here, now, we're going to have 4 m minus here, m is already out, so we have three, and we equate this to 0.

[00:03:17]Okay, so, now, we have m to be equal to 0 or 4 m - 3 to be equal to 0.

[00:03:30]Because we have to apply our zero product rule.

[00:03:34]Now, m is already 0 or from here, 4 m will be equal to 3.

[00:03:42]-3 becomes +3 on the other side.

[00:03:47]M here is 0 or from here, we have to divide both sides by four.

[00:03:53]This will now go. So, on that side on that side, our M will be equal to 3/4.

[00:04:06]Sorry, I wrote out of sight.

[00:04:09]So, to conclude, we have our M to be 0 or 3/4.

[00:04:16]But, the question now is are we really expecting two solutions for the equation?

[00:04:23]Let's apply the second method, and then we will know whether to agree with the two solutions or not.

[00:04:32]Okay, you have to stay for the second method because it's going to be faster and easier for you.

[00:04:40]Right?

[00:04:42]We have the square root of 3M over M squared.

[00:04:50]Okay, just M.

[00:04:52]Right? And it's equal to two.

[00:04:55]So, what do we do?

[00:04:57]Our first target now is to remove the square root.

[00:05:01]And to remove the square root, we have to square both sides of the equation.

[00:05:05]Yes.

[00:05:06]Without cross multiplying, let's square the both sides of the equation.

[00:05:11]So, we have square root of 3M over M the whole of this will be squared.

[00:05:19]And then, we have two squared on the other side.

[00:05:24]Right? I think this is very, very easy.

[00:05:27]And [snorts] from one of the laws of indices that we know if you have A over B all squared this can be written as A squared over B squared, right?

[00:05:42]Okay, so let me remove this.

[00:05:47]Okay, so the same thing will be applicable here now. We can write this as square root of 3 m squared over m squared.

[00:05:58]And this is equal to 4 on the other side.

[00:06:03]So, this can just take this out.

[00:06:07]And over there we have 3 m left.

[00:06:11]And it's over m squared.

[00:06:14]This is equal to 4.

[00:06:16]Now, we are not going to cross multiply.

[00:06:19]If we cross multiply, it will give us exactly what the first method gave us.

[00:06:24]So, we will not cross multiply, but we can do something here.

[00:06:29]From here we can write this as m.

[00:06:34]Okay, 3 into m over m squared.

[00:06:41]And this is equal to 4. So, from here now we can easily we can easily cancel out one of the m.

[00:06:50]Right? Because this has power of 1. And from one of the laws of indices, we pick one of the bases.

[00:06:58]So, we have 3.

[00:07:00]We pick one of the bases, which is m.

[00:07:02]Then we have 1 minus 2.

[00:07:05]And this is equal to 4, right?

[00:07:09]This is interesting.

[00:07:11]Very interesting. So, to continue, we have 3 m to the power of negative 1.

[00:07:21]Because 1 minus 2 is negative 1.

[00:07:25]As it is equal to 4.

[00:07:28]And from here again, we're looking for the value of m. So, we can easily take out this 3 from here as we divide 3 by 3 and divide 4 by 3.

[00:07:41]Right? So, this will enable us to have only m on the left-hand side.

[00:07:48]Okay, so we will continue from here so that we have m to the power of 1 to be equal to 4 over 3.

[00:07:59]But, what do you notice? We have m to the power of 1. From one of the laws of indices, too, this is 1 over m being equal to 4 over 3.

[00:08:10]And we're looking for m alone, so we'll have to take the reciprocal of this, which will give us m over 1. m over 1 is the same thing as m.

[00:08:19]And is equal to the reciprocal of this, which is 3 over 4.

[00:08:24]So, this is the value of m.

[00:08:27]And remember that the first solution, I mean the first method, gave us two solutions.

[00:08:34]We got m to be 0 or 3 over 4.

[00:08:39]But, the second method is giving us only one solution.

[00:08:42]Now, do you think the two solutions from the first method will satisfy the equation?

[00:08:49]Look at the equation again. Square root of 3m over m equals um 2.

[00:08:55]Okay, so the first um solution gave us um a solution which we have to reject because m cannot be equal to 0.

[00:09:07]Yes, if you put m to be 0, then you're going to have 0 over 0, and 0 over 0 cannot give you 2.

[00:09:17]As a matter of fact, it's undefined.

[00:09:20]So, to conclude to conclude this, m equals 0 will be rejected, and m equals 3 over 4 is the only solution.

[00:09:37]Thank you for watching.