Olympiad Mathematics | Indian | Can You Solve This? | The three solutions

[00:00:01]Hi, everyone.

[00:00:04]Do you think you can solve this one?

[00:00:09]We have the square root of X over five all to the power of three equals 125.

[00:00:22]Okay, and we are not just solving it. We want to solve this completely.

[00:00:28]What do you think we are going to do?

[00:00:31]Um, let's work on the term in the in the bracket, right?

[00:00:37]We can write that as X over five.

[00:00:41]This is to the power of two.

[00:00:44]The power of one over two rather.

[00:00:46]Because this is the same thing as the square root of one over five.

[00:00:50]The square root of X over five.

[00:00:53]Now, this is raised to the power of three.

[00:00:58]And it's equal to 125.

[00:01:04]Now, you know we can change the position of the power here.

[00:01:08]So that we can have X over five.

[00:01:11]This is raised to the power of three.

[00:01:14]And at the same time raised to the power of one over two.

[00:01:19]Okay, so what I have done is to change the position of the power.

[00:01:23]And this is equal to 125.

[00:01:28]Now, what I would like to do is to remove power. So to remove that, I have to square this.

[00:01:34]By multiplying the power by two, then I have to square the other side as well.

[00:01:40]So this will take this out.

[00:01:43]And now, on the left-hand side, we have X over five to the power of um three.

[00:01:51]Right? And this is equal to the square of 125.

[00:01:58]And that will give us 15,000 625.

[00:02:06]And this 16,000 15,625 is a perfect square.

[00:02:13]So, we have X over In fact, let's break this. This is X to the power of 3 over 5 to the power of 3, which will give us um Okay, let's rewrite it as 5 to the power of 3 because we can split the power.

[00:02:29]And it's equal to 15,625.

[00:02:34]And like I said, it is a perfect cube.

[00:02:37]Right? Since it's a perfect cube, let's express it in that form so that we have 25 to the power of what? 3.

[00:02:47]Okay, so from here, let's write X to the power of 3 to be alone on the left, and then here we have 5 to the power of 3 to multiply 25 to the power of 3. What we have done is um cross multiplication, right?

[00:03:07]Yes, we we have cross multiplied. And then this is X to the power of 3 being equal to 5 multiplied by 25. Then both of them will be raised to the power of 3.

[00:03:22]Okay, this is very very possible so that at the end of the day, we are going to have being equal to 125 to the power of 3.

[00:03:39]Yes, this is what we have, 125 to the power of 3. And then to solve it, bring this to the left, we have X to the power of 3 minus 125.

[00:03:51]And um we have power of three everything equals zero.

[00:03:56]Sorry, I wrote that out of sight.

[00:04:00]So, this is our difference of two squares.

[00:04:04]a cubed minus b cubed is equal to a minus b into a squared plus ab plus b squared.

[00:04:17]So, we're going to use this identity to solve the equation.

[00:04:22]a is going to be x, b is 125.

[00:04:28]Then, a is x.

[00:04:31]Then, ab that would be 125 multiplied by what? Multiplied by x.

[00:04:38]That is ab. Then, plus b b um squared that's going to be 125 uh squared.

[00:04:49]Right?

[00:04:50]Let's leave it like that and then we expand it. We equate it to zero.

[00:04:55]So, from here now, looking at this we have to apply zero product rule. So, it is either this is zero or the terms in the other bracket is equal to zero.

[00:05:10]Let's pick this one first.

[00:05:12]x minus 125 is equal to zero meaning that x is equal to 125.

[00:05:19]And this is our first solution.

[00:05:23]So, to get the other solution I am going to bring down the expression there x squared plus 125 x plus 125 squared.

[00:05:39]This is an expression, right? But, since we are using zero product rule, we will also equate it to zero.

[00:05:46]And it becomes a quadratic equation.

[00:05:49]Very quickly, we will bring out our quadratic general formula, which is - b plus or minus b squared - 4 * ac all over 2 * a.

[00:06:05]So, that to continue with this, we have x to be equal to In place of this - b, I'm going to put -125 because b is 125, c is 125 squared and a is 1. That is the coefficient of x squared.

[00:06:27]Plus or minus, we have b squared and that will be 100 That'll be 125.

[00:06:37]And there's a square on it, right?

[00:06:39]Then we have minus.

[00:06:42]There's a square on that.

[00:06:45]We have minus 4 multiplied by a is 1. So, even if you multiply by 1 it will not change anything. So, we multiply by c which is 125 squared.

[00:06:59]And this is all over 2 * 1, which will still give us 2.

[00:07:05]Now, this x is equal to -125 plus or minus we have the square root of 125 squared is common.

[00:07:17]So, write 125 squared as a common factor.

[00:07:23]125 squared divided by itself is 1.

[00:07:28]Okay, it's 1. Then minus this divided by itself, we're going to have only 4 on that side.

[00:07:38]So, we put our 4 and we divide by 2.

[00:07:42]To go on with what we have here, our X is going to be equal to minus 125 plus or minus we have the square root of 125 squared multiplied by minus three.

[00:08:00]1 minus 4 is 3, minus 3 rather.

[00:08:03]So, this is all over two.

[00:08:06]So, we continue from here.

[00:08:10]Okay, so from here now we have X to be minus 125 plus or minus I believe you know that square root of AB is the same thing as square root of A times square root of B.

[00:08:26]You can always split it in this way.

[00:08:30]So, we will now split what we have there.

[00:08:33]As we have square root of 125 squared multiplied by the square root of negative three.

[00:08:41]These will take this out. But mind you, this is over two.

[00:08:47]Okay, so if we go on with this, we're going to have X to be minus 125 plus or minus we already have 125 there multiplied by square root of negative three. So, we will work on that and get negative three multiplied by okay, positive three multiplied by negative one there. Both of them under the root.

[00:09:14]Then, we divide by two.

[00:09:17]Square root of negative one is I, so our X will now be minus 125 plus minus we write 125 times I from there then multiply this by root three.

[00:09:32]This is all over two.

[00:09:35]Now, look at one of the mistakes you will likely make. You might be tempted to add these two or subtract them. But, they are not the same. So, you have to leave it like this.

[00:09:47]Now, to bring down the three solutions now, we got X to be equal to 125 as our first solution.

[00:09:57]Then, X again is -125 plus or minus. No, we pick only the plus. Since we want to split the solution there.

[00:10:09]125 I then we have root three all over two.

[00:10:15]Then, the third solution is equal to -125 -1 25 I then we have root three.

[00:10:30]This is over what? Two.

[00:10:33]So, this this three are the solutions to the equation.

[00:10:38]Thank you for watching. See you in the next video.

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