Olympiad Mathematics | The complete solution | Indian Can You?

[00:00:00]Hi, everyone.

[00:00:04]You're welcome to Fico Mathematics.

[00:00:08]And today we have square root of 2 x over 2 raised to the power of 2 to be equal to We have from 2 over the square root of 2 x.

[00:00:28]Okay.

[00:00:30]This is it.

[00:00:32]And we have to understand that a over b to power 2 is the same thing as a squared over um b squared. Do you see that?

[00:00:46]So, [snorts] we're going to apply this to the left-hand side now.

[00:00:50]And that will give us the square root of 2 x squared over 2 squared.

[00:01:02]Okay, so we'll have this.

[00:01:04]And then on the right we have the same 2 over the square root of 2 x.

[00:01:11]So, that the square root and the square will cancel out.

[00:01:15]On the left-hand side we're going to have 2 x over 4 to be equal to 2 over the square root of 2 x.

[00:01:28]So, from here now, what do you think we are going to do?

[00:01:33]We can even cancel out, right? 2 will go here.

[00:01:38]We have 1, so we have 1 x. So, that 2 into 4 will give us 2.

[00:01:44]And this is equal to here we have 2 over what?

[00:01:49]The square root of 2 x.

[00:01:52]And at this point we are going to cross multiply so that this multiplying this will give us X root 2 X, right?

[00:02:03]And that will be equal to 4, which is 2 * 2.

[00:02:08]So, from here now, we just have to square both sides of the equation.

[00:02:12]So, we have X root 2 X the whole of this will be squared and it will be equal to 4.

[00:02:23]Okay, not just 4, but 4 squared as well.

[00:02:26]So, that's X to the power of 2 will come here since we know that the power will work for the two terms in the bracket. So, we have X squared and then this can now cancel this as we multiply by 2 X. So, this is equal to what? 16.

[00:02:47]Okay, so from this point we multiply these two together um 2 X to the power of 3.

[00:02:56]This is equal to 16.

[00:02:59]Now, we're going to divide both sides by 2.

[00:03:05]So, that this will go with this one and then X to the power of 3 is equal to 8, right?

[00:03:13]Now, let's solve this completely. So, we're going to have X to the power of 3 to be equal to 2 to the power of 3.

[00:03:22]Because 2 to the power of 3 will give us 8.

[00:03:25]Now, bring this to the left, we have X to the power of 3 minus 2 to the power of 3, which is now equal to 0.

[00:03:34]Okay.

[00:03:35]And I remember the difference of two squares difference of two cubes, a cubed minus b cubed and it's equal to a minus b multiplied by a squared plus ab plus b squared. Okay? This is our difference of two cubes.

[00:03:58]>> [snorts] >> And we are going to apply this to solve this problem.

[00:04:04]So, our a now is going to be x. So, we put x here minus b is two.

[00:04:12]Into a squared a is x squared plus ab that will be 2x.

[00:04:19]Then plus b squared which will be two squared and it's four.

[00:04:23]So, all of this is equal to zero.

[00:04:26]So, to continue with this we apply our zero product rule.

[00:04:31]So, either of this is going to be zero.

[00:04:34]So, x minus two is zero or x squared plus 2x plus four is equal to zero.

[00:04:45]So, if we continue we have x to be zero plus two.

[00:04:51]Uh that means that our x is two. This is one of the solutions.

[00:04:58]To get the other solutions, from here we have a quadratic equation. So, using quadratic formula a will be one. That is the coefficient of x squared.

[00:05:08]Then b is going to be two. That is the coefficient of x.

[00:05:13]And then um c is four which is the constant.

[00:05:18]>> [snorts] >> So, we'll now bring down the quadratic formula.

[00:05:22]If you can recall, we have x to be minus b plus minus then we have b squared minus 4ac.

[00:05:33]All of this is over two a.

[00:05:36]So, once you are able to recall the formula, you just substitute directly into it.

[00:05:41]So, that our x will be in place of minus b we are going to put minus two plus or minus b squared. That's going to be two squared.

[00:05:51]Then [snorts] minus four times one times four.

[00:05:55]Why? Because we have um a to be one and c to be four.

[00:06:01]So this is divided by two times one.

[00:06:07]Okay, so from this point x is minus two plus or minus we have four minus 16 and we divide by two.

[00:06:20]So that x will now be minus two plus or minus the square root of negative 12 over two. Four minus 16 is minus 12.

[00:06:31]Then if we continue from here, we're going to have x to be what? Minus two plus or minus we have the square root of 12 times the square root of negative one.

[00:06:42]This is all over two.

[00:06:45]So our x will now be minus two plus or minus the square root of four times two.

[00:06:55]Four times three will give us 12.

[00:06:58]The square root of negative one is i.

[00:07:01]This is all over two.

[00:07:04]So um to continue, we shall have x to be what? Minus two plus or minus square root of four is two and then multiply by this i, we have two i. Then we multiply by root three because three is not a perfect square.

[00:07:21]It will still be under the square root sign.

[00:07:24]So that we divide all of this by two.

[00:07:27]Now, the value of x will be equal to two into minus two is minus one.

[00:07:35]Two into two i, that is i, and we have root three.

[00:07:40]So this is a two-in-one kind of solution.

[00:07:43]We're going to bring the complete solutions together already.

[00:07:47]We got the first solution to be two.

[00:07:50]That's our X1.

[00:07:52]Then, the second solution is from here.

[00:07:55]-1 + I we have √3.

[00:08:00]So, that the last solution, which is the um from this part is -1 I and we have √3. So, these are the three solutions to the equation.

[00:08:15]Thank you for watching. See you in the next video.

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