Olympiad Mathematics | Indian| Can you solve this?

Indexé : 2026-05-22

[00:00:00]Okay, if you're ready let's provide a complete solution to what we have here.

[00:00:08]Solution.

[00:00:11]We have the square root of 8 over x squared to be equal to x over the square root of 8.

[00:00:22]Now, do not let this root confuse you, right?

[00:00:27]So, what should we do?

[00:00:29]Cross multiply So, we're having x to the power 2 * x. That will give x to the power of 3 and it will be equal to square root of 8 multiplied by the square root of 8.

[00:00:44]And then what do you do? Remember that if you have square root of a multiplied by the square root of b this is actually the square root of ab.

[00:00:57]Okay?

[00:00:58]So, if I do the same thing here now, we're going to have x to the power of 3 being equal to the square root of 8 * 8 and that is 64.

[00:01:11]You can see that, right?

[00:01:13]So, our x to the power of 3 will now be the same as the square root of 64 and that is 8.

[00:01:20]What do you observe?

[00:01:22]Okay, what just happened is that if you have square root of a * square root of a the answer is a.

[00:01:32]Square root of 2 * square root of 2, the answer is 2.

[00:01:36]Okay, so this is what just happened and from here we know that x to the power of 3 is going to be equal to 8 which is 2 to the power of 3.

[00:01:47]Now, we have the same powers but then let's bring everything to the same side.

[00:01:53]So, x to the power of 3 - 2 to the power of 3 is equal to 0.

[00:02:00]We can conclude that x is 2.

[00:02:03]But that will not give us the complete solution. So, let's break it down into detail.

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[00:02:18]Okay. Now, we know that a cubed - b cubed is an identity that is equal to a - b multiplied by a squared + ab + b squared.

[00:02:36]Now, relating to this, a is x and b is 2. So, we're going to have x - 2 into a squared, that's going to be x squared + ab, that'll be x * 2, and it's 2x.

[00:02:56]Then + b squared, which should be 2 squared.

[00:02:59]Right? 2 squared is what? 4.

[00:03:03]2 squared is 4. So, we close this as we equate to 0.

[00:03:10]We have to equate to 0 because of this, right?

[00:03:13]So, now from here, we carry out our zero product rule because since we're multiplying these two terms to get 0, either of them is going to be equal to 0.

[00:03:25]So, let's this be equal to 0 first. x - 2 is equal to 0.

[00:03:32]And from here, we are getting x to be 0 + 2.

[00:03:37]And our x is equal to 2, our first solution.

[00:03:42]Now, to get the other two solutions, we're going to bring what we have there.

[00:03:46]We're going to bring it down.

[00:03:48]We bring x squared + 2x + 4 = zero. So, we'll bring it down.

[00:03:56]Okay, so from here now, we have a quadratic equation and we are going to solve it using the quadratic formula. A is one, that's the coefficient of x squared.

[00:04:06]B is um B is two and C is four.

[00:04:13]The formula is x equals minus b plus minus the square root of b squared minus four a c all over two times a.

[00:04:25]So, now that we have our a, b, and c we will put them into this formula.

[00:04:31]So, our x will now be minus two plus minus b squared that's going to be two squared minus four times one times four.

[00:04:44]Because a is one, c is four.

[00:04:48]And we divide this by two times one.

[00:04:52]Okay, so from here our x is equal to minus two plus or minus we have two squared that is four minus four times one times four that is um 16.

[00:05:08]So, we divide this by two.

[00:05:11]And um to go on, we are going to have x to be minus two plus minus four minus 16 is minus 12.

[00:05:21]And this is all over two.

[00:05:24]Continue to get x to be minus two plus or minus square root of 12. Oh, I left out a negative, right? So, I can multiply this by square root of negative one.

[00:05:38]And I know that everything is still over two.

[00:05:42]Don't stop there.

[00:05:43]Our x is going to be equal to minus two plus or minus. Look at square root of 12 is the same as 4 * 3.

[00:05:55]Then the square root of negative one is imaginary.

[00:05:59]Everything here is over two.

[00:06:02]And by the way, we can split these two from one of the laws of indices.

[00:06:07]You know, um if you have the square root of AB, like I said before, it is square root of A * square root of what? B.

[00:06:16]Okay, so that means that we can find the square root of four separately and the square root of three separately.

[00:06:25]Now, our X will be minus two plus or minus. Square root of four is two.

[00:06:31]Multiply by this I, so we have two I.

[00:06:34]Then multiply by root three and we have root three.

[00:06:39]This is all over over two.

[00:06:43]We can split what we have to get X to be minus two over two plus or minus. We have two I root three over two as well.

[00:06:56]So, what do we have?

[00:06:59]We have our X to be minus two over two is minus one plus or minus. Two over two is one. One times I is I. Then we have root three.

[00:07:11]This is a two-in-one kind of solution because of the plus or minus.

[00:07:15]It's a matter of facts, this means that X is equal to minus one plus I root three or minus one minus I root three.

[00:07:28]Okay, so let's bring the three solutions together.

[00:07:32]Okay, so the equation we have solved is the square root of eight over X squared to be equal to x over the square root of 8, right? And the solutions are x x1 equals 2.

[00:07:51]That's the first solution.

[00:07:53]Then the second solution x2 is equal to -1 + i root 3.

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

[00:08:02]Then the third solution to the equation is -1 - i root 3. So, these three are the solutions to the equation.

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