Olympiad Mathematics | How I got the two solutions.

Added: 2026-05-09

[00:00:01]Okay, let's provide a solution to this one. Quick one.

[00:00:08]We have 2 x to the^ of 2 = 2 to the^ 3 + 4 to the power of 3 then + 6 to the^ of 3.

[00:00:24]So we are looking at the best way to deal with this.

[00:00:29]Um let's write 2 to the^ 2 x ^ 2 and it will be equal to 2 ^ 3 + 4 is 2 * 2 * 2 you know that is the same as 2 right? So 2 and we write um okay I think it's better we write 2 * 2. Yes. So write 2 * 2 to the^ of 3 then + 6 is going to be 2 * 3 and it is to the same power of 3.

[00:01:09]Now at this point what do we do? We'll apply one of the laws of indices that says m * n to the power of a is the same thing as m^ a * m okay * n rather to the same power of a. So we can split like this. So the same thing will happen to this two and uh we'll have 2x ^ 2 to be equal to 2 to the^ 3 plus here we have 2 ^ 3 * 2 to power 3 then plus on that side 2 ^ 3 * 3 to the^ three.

[00:02:07]Okay, if you're following then you're doing well.

[00:02:11]Now we have 2 x to the^ of 2 to be equal to Now look at these three terms. We have 2 ^ 3 2 ^ 3 2 ^ 3. So it comes out as a common factor. Here one will be left plus here we have another 2 to^ 3 plus here we have 3 to power 3.

[00:02:40]This is interesting right? So we have 2x ^ 2 being = 2 ^ 3 into now the addition okay this is going to give us 1 + 8 + um + 27 and at the end of the day we have 2 x^2 to be equal to 2 cub table multiply by 36 the addition and um let's do something here we can divide both sides of this equation by twoide by twoide by two two will take two out okay so if that happens then we'll have x to the^ 2 to be equal two. Now look at what will happen to this. This is having power of one, right? So we can clear out the fraction as we pick one of the bases which is 2. Then we have 3 - 1. So this is going to multiply 36.

[00:03:59]And what is 36? 6 to the power of 2. Now you will see why I wrote 36 in index form.

[00:04:12]Okay. So now we have our x ^ 2 to be equal to 2 to the^ of 2 then multiply by 6 to the power of 2. Now what again do we do from here?

[00:04:28]um from the law I just talked about that if you have m n right to the power of a it can be written as m^ a * n power a right so now we are going to write this in the form of this yes this will be written in the form of this so that we'll have the same power for both of them. So let's do that.

[00:05:06]Okay. So from here now we are going to write x² to be equal to 2 * 6 to the same power of 2. Okay. So we have x² to be equal to 2 * 6 12 then to the same power of two. Now we are having the same powers right? If you equate the bases directly you will not be doing it the right way because we need to have more than a solution from here. So what do we do? Bring this to the left and you have x^2 - 12^ 2 = 0.

[00:05:57]And this is our popular difference of two squares.

[00:06:01]And we know what it says that if you have a 2 - b 2. This is the same as a minus b * a + b.

[00:06:18]This is what we call the identity for difference of two squares. And in this case our a is x and our b is 12. So that means we are going to have x - 12 * x + 12 and this is all equal to zero.

[00:06:40]So that at this point we can apply our zero product rule. Since we are multiplying the two terms to get zero.

[00:06:48]So x - 12 is either 0 or x + 12 is 0.

[00:06:58]From the first case x will be 0 + 12.

[00:07:04]Meaning that x is = 12. This is one of the solutions.

[00:07:10]x = 12. And then from the second case over there, okay, we have x to be = 0 - 12. And this means that x is = -12.

[00:07:27]Right? So we have another solution here 2. And um to conclude we will say that x is = 12 or -2.

[00:07:43]Okay let's verify our work. The equation is 2 x^2 to be equal to 2 ^ 3 + um 4 ^ 3 + 6 to the same power 3. Now let's work with x to be 12.

[00:08:05]Okay. So if we put x to be 12 then we'll have 2 * 12 2 right.

[00:08:13]And um let's go to the other side. This is 8 + 64 + 6 ^ 3 is 26.

[00:08:24]Now the addition from here will give us 288.

[00:08:30]Right? Then on the left hand side we have 2 * 12^ 2 is 144.

[00:08:40]So you can see that 2 * 144 will give us 288.

[00:08:46]So that is also um it's satisfying. Then if you pick -12 as well, you know you're going to square it. -12 squared will still give you positive. So this means that both values x = 12 or -2 both of them are satisfying the equation. Thank you for watching.