Olympiad Mathematics | Japanese | Can You Solve This?

Indexé : 2026-05-21

[00:00:00]Hello everyone.

[00:00:02]Are you ready?

[00:00:05]If you are ready, let's solve this one now.

[00:00:08]B to power 3 plus B to power 2 equals 36.

[00:00:15]Now we want to solve this problem here using balancing method. What do write what we have on the right in the form of this.

[00:00:24]So, what do we do? We look at B to the power of 3. Remember B to the power of 3 is expected to be less than 36.

[00:00:34]Right? So, if that is true, then B to the power of 3 should be numbers like 20, 21, 23, 25, 26, 28, 27. Right? And not up to 36.

[00:00:49]But then, if we pick all these numbers I've mentioned, which one can be expressed in the power of 3?

[00:00:57]Which one?

[00:00:58]Is it not 27?

[00:01:00]Right? Okay, let's try that 27. So, we have B cubed plus B squared equals 27. So, 27 plus what will give us 36? It is 27 plus 9.

[00:01:16]In this case, we have our B to power 3 plus B to power 2 to be 3 to power 3 plus 3 to power 2. Now, we have balanced what we have on the left and what we have on the right-hand side.

[00:01:33]But, okay, we can even conclude that our B is equal to 3 from here.

[00:01:39]But since we are interested in getting the complete solution, we have to continue going. Now, B to the power of 3 minus 3 to the power of 3 will be collected together.

[00:01:50]Then we have plus open bracket, we have B to the power of 2 minus 3 to the power of 2 and everything equals 0.

[00:02:00]This is what I have done. This will come here to become negative. This will come here and it becomes negative as well.

[00:02:08]Now, we have two groups to simplify.

[00:02:11]The first is difference of two cubes and we know that x ^ 3 - y ^ 3 is an identity which is equal to x - y into x ^ 2 + xy + y ^ 2 and I bet you know, you know, I believe you know that this is difference of two squares and you're used to the identity.

[00:02:39]So, let's just continue.

[00:02:42]From here, our b our x is going to be b and our y is 3. So, x - y is b - 3.

[00:02:51]Okay. Then into x ^ 2 is b ^ 2 + xy is 3b because y x is b and y is 3. So, if you multiply, we get 3b. Then + b ^ 2 which is going to be um y ^ 2 rather which is going to be 3 ^ 2 and 3 ^ 2 is 9.

[00:03:18]Close this and we'll go over to the difference of two squares.

[00:03:22]For the difference of two squares, we have b - 3 b + 3 and everything here is equal to 0.

[00:03:32]So, if we go on from here, what do we do?

[00:03:36]We have b - 3 here and here. So, it is a common factor. Write it down.

[00:03:43]In here, we have this and there we have that. So, we're going to add this and this because of this sign.

[00:03:50]So, we have b squared plus 3b plus 9 plus b plus 3.

[00:03:58]plus b plus 3.

[00:04:01]So, we equate to 0.

[00:04:04]Now, our b minus 3 is still a common factor.

[00:04:09]Then here we have b squared. 3b plus b, that will give 4b.

[00:04:15]Then 9 plus 3, that will give 12.

[00:04:19]And this is all equal to 0.

[00:04:22]By the way, we are multiplying two terms here to get 0. The whole of this is considered a term and this is also considered a term. So, it's either this is equal to 0 or this is equal to 0.

[00:04:37]Okay, so we're going to express this and equate each of them to 0 and then solve it, right?

[00:04:46]Okay, so we are saying that b minus 3 is 0 or b squared plus 4b plus 12 is 0.

[00:04:58]From here, our b is equal to 0 plus 3.

[00:05:03]And the value of b is 3.

[00:05:05]This is one of the solutions.

[00:05:11]Now, to get the other solutions, we'll bring down the quadratic equation there, which is b squared plus 4b plus 12 equals 0.

[00:05:25]We are going to solve this quadratic equation using the formula method.

[00:05:29]Oh, the formula has a and b. Okay, it has a, b, c.

[00:05:34]So, because of that, permit me to rewrite this as x squared plus 4x plus 12 equals 0.

[00:05:45]So, that we can now use formula, x = - b plus minus we have the square root of b squared minus 4 ac all over 2a.

[00:05:58]Now, our abc here, we need to know the value of a.

[00:06:04]A is a coefficient of x squared, b is a coefficient of um x and c is a constant. Uh yes, c is a constant. So we're going to put them into this um this formula so that our x will be -4 plus or minus we have b squared which is going to be 4 squared minus 4 times 1 times 12.

[00:06:35]Okay, so this is all over 2 multiplied by 1.

[00:06:40]So that if we go on from here what do we have?

[00:06:45]Our x will be -4 plus or minus 4 squared is 16. 4 times 12 is 48.

[00:06:56]So we are dividing this by 2.

[00:07:00]We divide that by 2. So that if we go on we have x to be equal to -4 plus or minus the square root of 16 minus 48 is minus 32.

[00:07:17]This is all over over 2. Now, the next thing we do we will do is to try to you know, we will try to remove this negative so that we can have x to be -4 plus or minus the square root of 32 multiplied by the square root of negative 1.

[00:07:40]And all of this is over two.

[00:07:45]Okay, so from here now we get our x to be minus four plus or minus the square root of 32 is 16 times two.

[00:07:55]Then square root of negative one is imaginary.

[00:07:59]Everything is over two.

[00:08:01]So our x now is minus four plus or minus square root of 16 is four.

[00:08:08]Multiply by this i then we have root two coming in.

[00:08:13]All of this is over two.

[00:08:16]Now, what do we do? X will be equal to two into minus four is minus two.

[00:08:22]Two into four i that will be two i then we have root two.

[00:08:27]Now we have two in the one solution.

[00:08:31]Remember whatever we have here also stands for the b, right?

[00:08:37]Yes, it stands for the b.

[00:08:39]Now let's bring the three solutions together. We got to be the first solution to be three.

[00:08:48]Then the second solution b two is from here minus two plus two i root two. This is the second solution.

[00:08:59]And the third solution b three is minus two minus two i root two. So these are the three solutions. Thank you for watching. Consider subscribing to my channel for more.