Olympiad Mathematics | The two methods used

Added: 2026-05-06

[00:00:01]Hi everyone.

[00:00:04]Let's solve this problem here in two ways.

[00:00:10]Okay. So if you have issues solving sodic equation then you have to pay attention.

[00:00:19]We have the square root of B / 2 B and this is equal to 4.

[00:00:26]Like I said, I was going to solve this in two ways. Um, we have square root of B to be equal to 2 B * 4.

[00:00:39]Okay. So, what I'm doing now is called cross multiplication.

[00:00:44]So, our square root of B will now be 2 B * 4 is 8 B.

[00:00:52]Now, what do we do? Our target is to remove the negative um the square root because without removing the square root you can't find what is in there. Okay, especially now that it is an unknown variable.

[00:01:06]So we have the square root of b raised to the power of two and then we have 8 b raised to the power two. Now if I fail to put this in brackets I will have the wrong answer.

[00:01:23]Now the square root of the square will go. So B is alone and is equal to 8 B * by 8 B. This is what it means. Now let's multiply so that we can have B to B. A * 8 is 64. B * B is B 2.

[00:01:46]This is it. And we'll write the one with the highest power first.

[00:01:53]Let's write the one with the highest power and uh write 64 b².

[00:02:01]Then this will become b everything is equal to zero.

[00:02:06]Now we have a quadratic equation meaning that we are going to have two values of b.

[00:02:14]Let's factor out b. And here we have 64 B minus B into B is 1. This is equal to Z.

[00:02:25]And at this point we can apply our zero product rule which says that B is either zero or 64 B - 1 is 0.

[00:02:40]So to go on we have B already zero or 64 B is equal to 1 because this is -1 it also positive one on that particular side our B is already zero or from here I hope you know that we are dividing both sides by 64 so that means B will be alone and is equal to 1 / 64 four.

[00:03:10]Now to conclude from the first method we say that B is equal to 0 or 1 / 64. Now we do not know if both of them are going to satisfy for now. So let's leave this and apply the second method. Let's see what we're going to have.

[00:03:33]Okay. So if you're ready, let's apply the second method.

[00:03:42]Okay, we still have the square root of b / 2 b = 4.

[00:03:52]Remember the first method gave us a quadratic equation.

[00:03:56]So let's see what the second method will give us. I don't want it to give us quadratic. And this is what we'll do. We can use this two to multiply four. So we have the square root of b over b to be equal to 8 because 2 * 4 is 8.

[00:04:15]Can see that this will not give us quadratic equation. So what do we do from here? This is the same thing as b ^ 1 / 2 / b.

[00:04:28]Okay. Yes. because square root of b is b ^ 1 / 2 which is equal to 8.

[00:04:35]Now what do we do from here? Again we are going to understand that this is also having a power of one. So that if you use this um law of indices that says x to the power of b um let me use a okay x to the power of a divided by x to the power of c. This is the same thing as x to the power of a minus c. Because we have division in between them we subtract the powers. So we're going to do the same thing and we have um b to the^ 1 / 2 - 1 to be equal to 8.

[00:05:23]Now 1 / 2 - 1 you know what that will give right to give us -1 / 2. So this is equal to 8.

[00:05:34]If you want to remove that negative then you should have 1 / B to the^ 1 / 2 to be equal to 8. Now the negative has gone out. And then the next thing you can do is remember that this is over one. Let's take the reciprocal of both sides of the equation. So if we do that this will go up and it will be b^ 1 / 2 / 1 which is the same as b ^ 1 / 2 and it's equal to we turn this around as well. We have 1 / 8.

[00:06:12]I told you this will not give us any um quadratic equation.

[00:06:19]So now we have b to the^ 1 /2. How do we remove the power? To remove the power 1 / two, we will square both sides of the equation. So 1 / 8, this will be squared as well.

[00:06:34]Now two can cancel itself from here.

[00:06:39]And what does it mean? It means that b is alone and is equal to 1 8 * 1 / 8 because of the square on it. And what would this give us? This will give us that our b is = 1 / 1 * 1 is 1. 8 * 8 is 64. So we have b to be = 1 / 64. Like I said, we will not be having quadratic equation and that's why we are having just one solution.

[00:07:12]Now let's go back to the equation given to us cuz we have to put this back into the equation.

[00:07:23]Okay, so this is the given equation and the first method gave us B 0 or 1 / 64. Now if you put B to be zero, you're never going to have four on the other side because you're going to have 0 / 0 which will not give you four. Okay? So this means that B= 0 will be rejected.

[00:07:51]Now let's put in b to the 1 / 64. That means we have<unk> 1 / 64 over 2 * 1 / 64.

[00:08:03]Yes, cuz we have to be under. Now this implies that we have square root of 1 is 1, square root of 64 is 8. Then 2 into itself is 1, 2 into 64 is 32. So we having this to divide 1 / 32 by the way. This is division.

[00:08:26]This is division.

[00:08:29]Okay. Oh yes I think we can see it now.

[00:08:32]So if that is division then it means that we have 1 / 8 * 32 over 1 and at the end 8 into itself is 1. 8 into 32 is 4. So the answer is 4 32 and 4 32 will still give us what? Sorry, 4 1 will still give us 4. No wonder the given equation is the square root of b / 2 b = 4. So we are very correct to say that our b is = 1 / 64.

[00:09:11]So it is only the second method that give us what we need completely without giving us an extraneous um solution.