Olympiad Mathematics | Russian | Can You Solve This?

Ajouté : 2026-05-13

[00:00:01]Okay, let's solve this one.

[00:00:05]Quick one.

[00:00:07]X over 3 * X / 3 = 3 /X.

[00:00:15]This is a quick one. We want to provide the complete solution.

[00:00:20]X * X 2 3 * 3 3 2 R. So I have 3^ 2 here and it's equal to 3 over x.

[00:00:33]So what again do I do? Cross multiply.

[00:00:38]Yes, we we just have to cross multiply.

[00:00:42]And um to do that we shall have x to the^ 3. x² * x is x ^ 3. and then 3 2 * 3 is 3 to the power of 3. So we have um the same powers but different bases. Now if we say x is equal to 3 we will be correct but that will not provide the complete solution or produce the complete solution to the equation and that is the more reason we're going to bring this to the left as we have 3 to power 3 and all of this is equal to zero.

[00:01:26]Now we have our difference of two cubes.

[00:01:29]Okay, difference of two cubes. So when you have two terms, you know, to the power of three and you're taking their difference just like this.

[00:01:40]Imagine we have a cube minus b cube.

[00:01:44]This is what we call an identity because once you have this, you already know what it is. A minus b.

[00:01:53]Then we have a squ + a b + b 2. Okay. So this is an identity in mathematics.

[00:02:05]So that from here our a is going to be x and the b is going to be three. So we put them into this identity and we will have x - 3 in place of a minus b. Then a 2 is x 2 + a b that's going to be um what is a? Our a is x. So x * 3 that is 3 x + b 2 which is 3 2 and 3 2 is 9. and all of this is equal to zero.

[00:02:47]So at this point what do we do? Our zero product rule. Okay, our zero product rule.

[00:02:57]I believe by now you know when to apply this rule. We apply this rule when we multiply two terms to get zero. For example, if you have P * Q and it's equal to 0. So you're saying that it's either one of this is zero. If not, you cannot have zero on the other hand.

[00:03:21]P * 0 is 0. 0 * Q is 0. So either of them is zero. And that's the same idea we're applying over here. Let's get it on.

[00:03:33]So we have our x - 3 to be zero or on the other hand x^2 + 3x + 9 is = 0.

[00:03:49]And on this side x is just 0 + 3 and we have our x to be equal to 3. This is one of the solutions.

[00:04:05]And to get the two um two more solutions from here, we will just work on this um on this um equation because this is already a quadratic equation. Working on this will um give us two more solutions.

[00:04:21]So we're going to have three complete solutions. Let's pick this one.

[00:04:28]Okay. So here we are. From here we are getting two more solutions from here. So the first thing is to get our ABC since we're going to use quadratic formula for this. A is a coefficient of x² 1. B is a coefficient of x 3 and c is a constant which is 9. So what is the quadratic general formula? Quadratic general formula is x = - b + or minus b^ 2 - 4 a c over 2 a. But mind you the x we have here is because we are looking for the value of x in the equation. If you're looking for the value of um y here should have been y. Okay. So let's substitute and our x will be equal to -3 because b is 3. Then b² that's going to be 3^ 2. Then we have -4 * 1 cuz a is 1 and our c is 9. Take note of that. This is all over 2 * 1 and 2 * 1 will still give us 2. So to go on with what we have our x will be - 3 + or minus 3^ 2 is 9 and then 4 * 9 is 36.

[00:06:01]So this is 36.

[00:06:04]All of this is over 2 * 1 and that is 2.

[00:06:09]And here is a confusion for many of my students. They'll be tempted to take the square root of 9 and then take the square root of 36 separately, but that will give you a wrong answer.

[00:06:23]So what do you do? 9 - 36 before you take the square root. So our x will be - 3 + or minus we have the square. Okay.

[00:06:36]Can even reduce it since it's going to be -27.

[00:06:40]Yes. 9 - 36 is -27 and this is over 2.

[00:06:46]Now let's work on this - 27.

[00:06:49]Yes, our x is going to be - um 3 + or minus<unk> 27 * roo<unk> of -1 all over 2. This multiply by this will give this back.

[00:07:05]And at this point our x will be - 3 + or minus the square root of look at 27.

[00:07:14]27 has a perfect square factor of 9.

[00:07:19]Then multiply by 3. That's 27. So square root of -1 is i. So we multiply this by by i. Right? And everything is going to be over two. And from here we have our x to be equal to - 3 + or minus<unk> 9. You know you can split this. You find square root of 9 first.

[00:07:47]Then square root of 3. Square root of 9 will give us 3. Multiply that 3 by i.

[00:07:52]That will three. That will be 3 i then this roo<unk>3 will come down. And this is all over over two. Okay, I think at this point we can stop. Mind you, this is a two in one solution because of the plus or the plus or minus. So I would like us to bring the three solutions together before. Now we got our x to be equal to 3. That will be the first solution. Then the second solution will come from here where we can have -3.

[00:08:34]Okay. X2 is - 3 then + 3 I <unk>3 and everything here is over two. This is because we have just picked the positive. So we're going to pick the negative again. So our x3 now that is the third solution is - 3 - 3 i then we have <unk>3 and everything is over two. So these are the three solutions to the given equation. Thank you for watching. If you have not subscribed, please do.