Secret tricks in solving Olympiad radical equations

Indexed: 2026-05-17

[00:00:00]Let me show you how simple this radical equation is. And it is very simple to comprehend.

[00:00:06]All right, so we are we shall use what we call substitution method to solve this.

[00:00:13]Okay. Now, the equation says the square root of 39 minus X plus the square root of 7 minus X is equal to 8. So, you're asked to find the value of X.

[00:00:24]Now, I want the whole of this. Let me say let the whole of this be M.

[00:00:30]Then the whole of this be be N. Okay.

[00:00:34]So, if I say the whole of this be M, okay, and the whole of this be N. Of course, this is plus which is equal to 8.

[00:00:44]Okay, so we shall have M plus N is equal to 8.

[00:00:49]Now, do you also believe that M squared minus N squared is equal to 32? Let me prove this.

[00:00:58]Let me prove this to you. Of course, this is written in a difference of two squares and we shall have M plus N multiplied by M minus N to be equal to this 32 can be written as 8 times 4.

[00:01:14]Okay.

[00:01:16]8 times 4.

[00:01:18]Now, this is 8. That means let's equate this to 8 and equate the the one of minus to 4 because this 4 is smaller. So, we shall have M plus N will be equal to 8.

[00:01:29]And also M minus N is equal to 4. Wow. So, our M plus N is equal to 8 from this place.

[00:01:39]So, that means M squared minus N squared is equal to 32. That means the statement holds true.

[00:01:46]Now, how can we solve this?

[00:01:49]Now, we shall solve this by adding equation one to equation two.

[00:01:53]Of course, I can title this one equation one and that is that this one equation two.

[00:02:00]Now, I'm going to add equation one to equation two. Of course, m + m will give us 2 m.

[00:02:06]+ n + - n that one has gone which is equal to 8 + 4 is 12.

[00:02:16]To find the value of m, divide both sides by two and divide here by two.

[00:02:20]This cancel out. This implies that our value of m will be equal to 12 / 2 is 6.

[00:02:26]That is the value of m. Let's find the value of n.

[00:02:30]Now, we notice that m - n is equal to four, but our m is six.

[00:02:37]From here, we can see that 6 - n is equal to four, which means that six when this cross over we shall have minus four, which is equal to minus cross over we shall have n. Of course, this implies that n is equal to 6 - 4 is 2.

[00:02:53]So, our m is 6 while our n is 2.

[00:02:57]So, if that is true, remember that m is equal to this while n is equal to this. So, we shall have m we shall have m to be equal to the square root of 39 minus x.

[00:03:14]But, our m is six.

[00:03:16]So, if our m is six, that means we place this with six.

[00:03:19]Okay?

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[00:03:26]Now, what do I do to clear this radical?

[00:03:28]I square both sides.

[00:03:30]If I square this side, I will also what?

[00:03:32]Square this side. Of course, this will cancel out this.

[00:03:35]This side becomes um 36. Please don't mind me because of space, I need to do it do it like that.

[00:03:42]Now, this become 36, okay? To be equal to 39 minus x, which means 36 when this cross over we shall have minus um 39 to be equal to minus x. That means minus x will be equal to minus three.

[00:04:02]Which means if you clean up this minus in both sides, that means x is equal to three.

[00:04:06]That is for the value for m to be equal to this. So, let let's prove the value of n whether we are still going to get x to be equal to three.

[00:04:16]So, we notice that we know that um our n, which is two, is equal to the whole of this, which is the square root of seven minus x.

[00:04:30]So, if you square both sides, you square this side and you also square this side.

[00:04:34]This cancels out this, we shall have two squared is four, which is equal to seven minus x.

[00:04:40]Of course, four plus um plus seven crosses over, we shall have minus seven to be equal to minus x.

[00:04:47]Four minus seven is minus three to be equal to minus x.

[00:04:51]So, therefore, the value of x is equal to three because when this cancels out, this x is equal to three. So, for m and m is true for this solution. So, therefore, x is equal to three. Thank you for watching. Share this video, follow us, and subscribe for more math tips like this. Turn on your notifications so that next time I upload my videos, you'll see most of them.