Olympiad Mathematics | Indian | Can You Solve This One?

[00:00:01]Hi, everyone.

[00:00:04]If you are ready, let's provide a complete solution to this equation here. We have 3x to the power of 4 equals x plus 8 raised to the power of 4.

[00:00:25]Now, from what we have already, we are expecting four different solutions.

[00:00:31]Four different solutions, right?

[00:00:33]So, let's see how we're going to get them.

[00:00:36]Now, the first thing I will do is to break the powers so that I can have 3x to the power of 2 and then whatever I have, I will square it again.

[00:00:50]Okay, I've divided the power into two.

[00:00:53]I have two and then two, but the the relationship between the two powers is multiplication.

[00:01:00]Then, here I have x plus 8.

[00:01:05]This again will be raised to power two and then the whole thing is raised to the power of two again.

[00:01:14]So, that's from here we can expand what we have inside of the bracket. So, the expansion of 3x squared will give us 9x to the power of 2.

[00:01:26]Remember, the power outside will still come out.

[00:01:30]Okay?

[00:01:31]Then, here the expansion of this what will it give us? Remember, we use this identity to carry out that expansion.

[00:01:40]Let's [snorts] say we have a plus b and it's raised to power two.

[00:01:44]This is the same thing as a squared plus 2ab plus b squared. This is the identity.

[00:01:53]So, applying this identity to expand this our a is x. So, here I'm going to have X squared plus 2 * A * B is going to be 2 * X * 8.

[00:02:06]And that will be 16 X.

[00:02:10]Then we have B squared, which is going to be 8 squared. 8 squared is 64.

[00:02:15]So, there we have 64.

[00:02:18]Remember that the square outside will still come down here.

[00:02:24]So, I'm going to remove what we have right here.

[00:02:30]So, from here now, we are going to bring this to the left.

[00:02:37]So, we have 9 X squared to the power of two minus open bracket, we have X squared plus um we have 16 X plus 64.

[00:02:55]So, this is squared. Everything is now equal to zero because I've moved everything from the right to the left.

[00:03:04]And from this point, we are going to apply difference of two squares, right?

[00:03:10]And I believe you know that. We have A minus A squared minus B squared to be equal to A plus B * A minus what? B.

[00:03:24]So, this is the difference of two squares. So, we're going to work with the first um condition, the first case where we have A plus B. Our A is 9 X squared.

[00:03:35]So, we have 9 X squared plus B, which is the whole of this X squared plus 16 X plus 64.

[00:03:48]So, okay. Let's work on this first. This is going to be equal to zero. And then, when I come back um to this point, I will still equate it to zero, right? So, this is our first case for now.

[00:04:04]Um this plus this will give us 10 10 x squared plus 16 x plus 64 this is equal to zero.

[00:04:17]Okay, we can reduce what we have already, so that we can have 5 x squared plus here we have 8 x plus here we have 32 this is equal to zero. So, what I've done is to divide all through by two.

[00:04:36]Okay, so we have a quadratic equation.

[00:04:41]A is five B is eight and C is 32.

[00:04:50]And then, we use quadratic formula, which is x equals minus b plus or minus we have b squared minus 4 ac everything is over 2 * a.

[00:05:04]So, what we'll do now is direct substitution so that x will be equal to minus eight plus or minus we have eight squared, which is 64 then we have minus 4 * a let me do that here.

[00:05:20]Um c is 32 times um a a is five, right? So, a is five and then 5 * 4 will be 20, so we just have 20 here.

[00:05:35]Okay, remember this is um 4 * 5, which is 20, then times 32. That's how I got 20 * 32 here.

[00:05:44]Multiply, this is 0 0 this is 4 6.

[00:05:50]Okay, and we are having 640.

[00:05:54]So, this is what will be in the square roots.

[00:06:00]So, here I'll write 640.

[00:06:04]So, let me remove this.

[00:06:08]And it is all over two multiplied by five. That will be 10.

[00:06:13]So, we have to continue from here.

[00:06:17]Okay, [snorts] so from this points we have X to be what? -8 plus or minus we have the square root of this minus this is -576.

[00:06:33]And we are dividing by 10.

[00:06:35]So, from this points we have X to be what?

[00:06:40]Um -8 plus or minus the square root of 576 then multiply by the square root of -1.

[00:06:51]Okay, I have picked out the negative from there.

[00:06:55]And we divide this by 10.

[00:06:57]Now, our X is equal to -8 plus or minus the square root of 576 is um 24 multiplied by the square root of -1 and that is going to be I.

[00:07:12]So, this is all over over over 10.

[00:07:16]Okay, so we can reduce this as well.

[00:07:19]So, reducing this will give us X to be equal to um two into that is -4 plus or minus two into that is um 12 I and two into 10 is five. So, we have this in two places because it's a two-in-one solution.

[00:07:37]Now, we will not stop here, right? Let's go back to the other factor which I left out.

[00:07:44]Okay, I believe you remember this point.

[00:07:47]And we used this one. We are yet to use this. So, we will now say that 9 x squared minus the whole of this, right?

[00:07:58]Okay, but this negative will affect all the signs here. So, we're going to have x squared minus 16 x then minus 64.

[00:08:10]This is equal to zero.

[00:08:12]So, from here we are expected to have two more solutions.

[00:08:16]This minus this is 8 x squared minus here we have 16 x and then here we have 64 which is equal to zero.

[00:08:28]You know we can divide all through by eight. So, that here we have just x squared minus here we have two x and here we have eight.

[00:08:39]This is equal to zero.

[00:08:41]Now, we want to solve this to get the solution. This is a quadratic equation now.

[00:08:47]Okay, so from here you know we can use um um factorization method to solve this one here.

[00:08:54]Because we know that um minus eight is the same thing as minus four times two.

[00:09:02]Then minus four plus two will give us minus two x. So, what are the two factors now?

[00:09:08]The factors will be x minus four multiplying x plus two.

[00:09:15]So, this will be equal to zero.

[00:09:18]And um from here we use zero product rule again so that x minus four is either zero or x plus two is zero.

[00:09:30]So, from this part x is four.

[00:09:33]And then from here x is minus two if we collect terms.

[00:09:38]So, these two now are the solutions from these parts.

[00:09:43]We're going to bring the four solutions together.

[00:09:47]Let's call X1 to be four, then our X2 to be minus two.

[00:09:53]Then X3 is what we had from the first part. I think I wrote it down.

[00:10:00]It is minus four plus 12i divided by five. Then the third the fourth one, X4 is um Okay.

[00:10:14]Let me write the fourth solution, and it's going to be minus four minus 12i divided by five. So, these are the four solutions to the given equation.