Olympiad Mathematics | Japanese | Can You Solve This One?

追加: 2026-05-25

[00:00:00]Hi.

[00:00:02]Let's solve this problem here completely.

[00:00:06]We have the square root of two over X to be equal to X over two.

[00:00:13]We want to solve this completely. So, what do we do first?

[00:00:17]We have to remove the square root.

[00:00:21]And we do that by squaring it, right?

[00:00:24]Then we have X over two, which will also be squared.

[00:00:30]Yes, we have to square both sides of the equation. Now, this will take this out.

[00:00:35]So, we have two over X on the left.

[00:00:38]And on the right on the right-hand side, we have what we have there.

[00:00:43]X over two times X over two.

[00:00:47]So, we are going to have um two over X, which is equal to X squared over over two squared.

[00:00:57]Okay, if we have to write it that way.

[00:01:00]X squared over two squared. Let me write this well.

[00:01:04]This is two over X, right? So, that the next thing we can do is to cross multiply as we have X multiplied by X squared to be equal to two multiplied by two squared.

[00:01:19]By the way, if you look at this very well, you can conclude that the value of X is two.

[00:01:26]The value of X is going to be two. But, we have to solve this completely.

[00:01:31]Whether the real and the complex solutions, we want to get them.

[00:01:36]So, we continue. X multiplied by X squared is X cubed.

[00:01:41]Then two multiplied by two squared is two cubed.

[00:01:45]So, bring two cubed to the left.

[00:01:49]X to the power of three minus two to the power of three is zero.

[00:01:54]And what we have right there is our popular difference of two cubes.

[00:02:01]So, we know the identity already.

[00:02:04]a cubed minus b cubed is just the same as a minus b and then we multiply by a squared plus ab plus b squared.

[00:02:19]Right? You know about this, right?

[00:02:23]Okay, so using this identity now, our a minus b will be x minus 2.

[00:02:31]a squared is going to be x squared.

[00:02:35]Then plus ab is going to be x times 2.

[00:02:39]Cuz a is x and b is 2. x times 2 is 2x.

[00:02:43]Plus 2 squared is b squared is going to be 2 squared.

[00:02:47]2 squared is 4.

[00:02:49]And then we are equating to zero.

[00:02:52]So, from here we apply this rule.

[00:03:01]We apply this rule, which we call um zero product rule.

[00:03:05]We apply this every time we multiply two terms to get zero.

[00:03:10]Imagine you have p times q to be zero.

[00:03:14]How will it be zero? If either p is not zero or q is not zero.

[00:03:20]So, one of them must be zero for you to get zero here. The same thing is what we are going to do to this um equation there.

[00:03:31]Okay, so applying the zero product rule x minus 2 is zero or x squared plus 2x plus 4 will give us um zero.

[00:03:47]From the left hand side, x is going to be zero plus 2.

[00:03:51]And the value of x is just two.

[00:03:55]This is our real solution and um the first solution.

[00:04:00]Now, to get the other two solutions, we will bring down the quadratic equation right there, which is x squared plus two x plus four equals zero.

[00:04:15]And this quadratic equation has a to be one, has b to be two and it has c to be four.

[00:04:25]A is the coefficient here, b is this two, c is that four.

[00:04:30]And um what is the quadratic formula that we're going to use?

[00:04:34]It is x equals minus b plus minus we have b squared minus four ac and we divide this by two a.

[00:04:45]So, the most, you know, difficult thing to do here is remembering the formula.

[00:04:52]Once you can remember the formula, it's simple.

[00:04:56]Now, putting the the value of abc, we'll have minus two here, plus minus b squared is our two squared now, then four times a times four cuz a is one, c is four.

[00:05:15]So, we're dividing this by two multiplied by one since our a is equal to one.

[00:05:23]Then, the value of x will be minus two plus minus two squared is four.

[00:05:30]Four times four is 16.

[00:05:33]And we divide by two.

[00:05:36]We will not stop here and we will not find a square root of four and square root of 16 separately.

[00:05:45]So, we have to combine, we have to simplify before we find the square root.

[00:05:49]So, X will now be minus two plus minus square root of four minus 16 is minus 12.

[00:05:57]And we're dividing this by two.

[00:06:01]Okay.

[00:06:02]And um let's do this. We have X to be minus two plus minus square root of 12 then multiply by square root of negative one.

[00:06:12]This is um possible.

[00:06:15]This is possible because if you have square root of A times the square root of B, this is the square root of AB, right?

[00:06:25]So, that means square root of 12 times square root of negative one, it will be square root of negative 12. So, that is okay.

[00:06:34]But, remember that we're dividing this.

[00:06:37]We have to divide the whole of this by two.

[00:06:41]Now, look at your 12 there. We can um simplify that 12. So, X will be minus two plus or minus square root of four times three then square root of negative one.

[00:06:56]The whole of this is still over two.

[00:07:03]Okay, so from here now, we're going to get X to be minus two plus or minus we have the square root of four, which is two then square root of negative one is I, so we're multiplying this by I then multiply by root three.

[00:07:21]Three is still there because it's not a perfect square.

[00:07:25]And we divide this by two.

[00:07:27]So, we can go on to get X to be equal to two into minus two is minus one plus or minus two into two I is I then we have root three.

[00:07:42]Okay. Now, let's bring um um the complete solution together, right?

[00:07:49]Let's bring the complete solution together. We got x before now to be equal to two, our first solution.

[00:07:57]Then, we got x to be from here -1 + i root three.

[00:08:03]This is our second solution.

[00:08:06]And um the third solution is -1 - i and we have root three. So, these uh the three solutions to the equation.

[00:08:19]Thank you for watching.