Olympiad Tricks | Germany | Can You Solve This?

Added: 2026-05-06

[00:00:01]Okay, if you're ready, let's provide the complete solution to this problem right here.

[00:00:12]We have the square root of x divided by 2x equals 2.

[00:00:21]Okay, so from here, I want to bring out the complete solution.

[00:00:25]So, the first step I would take is to cross multiply cuz we know this is over one.

[00:00:32]Okay?

[00:00:33]So, root x multiplied by one will give us root x.

[00:00:39]And that will be equal to 2x multiplied by 2.

[00:00:45]So, this will be the square root of x to be equal to 4x.

[00:00:53]And um at this point, what do we do?

[00:00:57]We need to remove this square root from here.

[00:01:00]And the only way we can do that is to square it.

[00:01:04]And if you square the left-hand side, you will have to square the right-hand side. So, we'll have 4x squared.

[00:01:15]Now, if you do not put this square root um this bracket, it means that the square is for only the x.

[00:01:22]So, you have to put everything um in bracket. Now, this will take this out. So, x will be equal to 4x squared.

[00:01:34]And this can be split to get 4 squared times x squared. Okay? Take note of that.

[00:01:42]And now, we have our x to be equal to 16 multiplied by x squared.

[00:01:48]And that is 16x squared.

[00:01:52]The next point is that we write the one with the higher power first, and that is 16x squared.

[00:01:59]So, 16x squared is equal to x. Now, bring this x to the left.

[00:02:06]Okay?

[00:02:07]So, if we do that, we're going to have 16x squared.

[00:02:12]Then, we have minus x. Now, there's nothing on the right-hand side.

[00:02:17]So, the only thing we can do is to factorize this um equation. Mind you, this is a quadratic equation. So, we'll be expecting two answers, two values of x.

[00:02:30]X will come out. Here, we have 1x and 16, that is 16x.

[00:02:36]Okay, then minus x divided by x is one.

[00:02:40]And this is all equal to zero.

[00:02:43]So, we have to apply our zero product rule.

[00:02:47]Do not forget when to apply this.

[00:02:50]When you multiply two terms to get zero, you can apply this. But if this is not zero, let's say this is equal to two or one, you cannot apply what I'm about to apply. Okay? That's why they call it zero product rule.

[00:03:05]So now, it's either x is zero or 16x minus one is zero.

[00:03:14]So that if we go on, our x remains zero.

[00:03:19]From here, 16x will be equal to zero plus one, and that is one.

[00:03:26]So, what do we do?

[00:03:27]We are going to divide both sides of this by 16, right?

[00:03:33]Divide by 16 so that 16 can go.

[00:03:36]And now, what is the value of x?

[00:03:39]We have our x from here to be zero, or from here, we have 1 over 16.

[00:03:47]So, at this point, we have the solution.

[00:03:51]But the question is, do you think both of them are going to satisfy the equation?

[00:03:57]As a matter of fact, let's look at another approach.

[00:04:00]Yes, I believe that is going to be better and faster than this method.

[00:04:04]Let's try it out.

[00:04:07]Okay, so second method.

[00:04:12]If you do not understand the first method, just look at this one here.

[00:04:16]Um what we'll do is from here, we can write that as 1 over 2 multiplied by root x over x.

[00:04:30]This is equal to 2.

[00:04:33]Very simple.

[00:04:36]And again, we can even multiply two by two.

[00:04:40]So, if we do that now, everything that has to do with x will be on the left.

[00:04:45]So, we have square root of x over x to be equal to 2 * 2, that is 4.

[00:04:52]Interesting, right?

[00:04:54]So, the next point is that if you have m in fact, root m is same thing as m to power 1 over 2.

[00:05:03]Okay, so I'll write x in this form now.

[00:05:07]So, it's going to be x to power 1 over 2, then divide by x. This is equal to 4.

[00:05:16]And um from here, I am going to apply a law, one of the laws of indices, because I know this is to power one.

[00:05:26]And the law says if you're dividing and you have the same base like this, you can pick one of the bases, that will be x, then the power will be subtracted.

[00:05:37]We have 1 over 2 minus the power from the denominator, which is one.

[00:05:43]And everything is equal to four.

[00:05:46]I told you this was um going to be very fast and interesting.

[00:05:51]Now, we have 1 minus one um 1 over 2 minus 1. This is x.

[00:05:57]Okay, if you do not know how to work on this, just This is over one, right? So, the LCM of two and one is two.

[00:06:05]Then, two 2 divided by 2 is 1 * 1, we have 1 minus 2 divided by 1 is 2 * 1 is 2.

[00:06:13]This is equal to four.

[00:06:15]Okay, I'm trying to show all the steps.

[00:06:20]And um at this point, we have x to the power of minus 1 over 2, which is equal to four.

[00:06:30]Okay, so we will not stop here. We need to get this power away. Now, the first thing is to remove this negative.

[00:06:37]And to do that, we have one over everything, which is x to the power of 1 over 2. But the negative is gone, and we have four.

[00:06:48]Do you know that four here can be 4 over 1?

[00:06:51]Okay? So now, let's take the reciprocal of this. And the reciprocal will be x.

[00:06:59]Okay? The reciprocal will be x to the power of 1 over 2 over 1. And it will not change anything, so no need to bring the one under. Then, the reciprocal of this is 1 over 4.

[00:07:11]Mind you, we want to remove the power.

[00:07:14]Okay?

[00:07:16]Okay, so to remove the power down, we have x to the power of 1 over 2, we have to square it.

[00:07:24]And then, we square the other side. Oh, by the way, this was 1 over 4.

[00:07:29]So, write 1 over 4, 1 over 4.

[00:07:32]So, this can take this out.

[00:07:37]So, we have our x alone.

[00:07:39]Remember, it should be x to the power of one, but there's no point writing the one. Okay?

[00:07:45]So, this is equal to one over four.

[00:07:49]And then, there's a square on this as well. Square this side, you have to square the other side. So, this is 1 over 4 as well.

[00:07:59]And um at the end of the day, our x is 1 over 16. Now, what do you observe?

[00:08:06]The first method gave two solutions, which is x equals zero or 1 over 16. But the second method gave us just one solution.

[00:08:18]And if you want to verify, the first um method gave one solution that is not needed, and that is x equals zero.

[00:08:28]Because the equation is root x over 2x equals 2.

[00:08:34]And if you put x to be zero, we are having square root of zero over 2 * 0.

[00:08:42]Okay, at the end of the day, we have zero over zero, which cannot and will not give us two.

[00:08:48]So, x to be zero will be rejected.

[00:08:55]And x to be equal to 1 over 2 will be accept um 1 over 16, rather.

[00:09:04]1 over 16 will be accepted.

[00:09:08]Thank you for watching.