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

[00:00:01]Hi everyone.

[00:00:04]How do you solve this problem here completely?

[00:00:09]Meaning that you should produce all the solution.

[00:00:13]This is x ^ 4 + x ^ 2 / x = 10.

[00:00:21]Now from here x ^ 4 + x ^ 2 / x is = 10. We can cross multiply so that we have x ^ 4 + x ^ 2 to be equal to 10 what? 10 x.

[00:00:42]And you know what we always say right that you do not divide by variable. So what do you think I should do? Bring this to the left and we have 10 x ^ 4 + x ^ 2 - 10 x to be equal to zero.

[00:01:02]So the next step to take is that we look at all the terms on the left and all of them have x. So you're going to bring out your x as the common factor. So that here there's x ^ 3 remaining and here x remains. Here we have 10 then everything equals zero.

[00:01:24]And from this point you can gladly apply your zero product rule so that you can say that um either x is zero or x cub - x - 10 is 0. So once we've said this we have a solution already. We will come back to address this solution here. And from here we are expecting three solutions.

[00:01:52]Let me bring it down here. Okay. I think this is plus. Yes, it's addition from the equation, right? So here we have x cub + x - 10 = 0.

[00:02:10]So the next step for us to take is that we look at the 10 here, right? We look at the 10 and we know we can break it into 8 + 2.

[00:02:24]This is because I know that 8 is the same as 2 ^ 3 and this two will represent x.

[00:02:33]So we have balanced what we have. Now this is x to the^ of 3 + x - open bracket 8 is 2 ^ 3 + 2 and this is equal to zero.

[00:02:50]The next step is that we open the bracket and reposition them so that x cube and 2 cube will come together. Then we have x this negative will affect this and it turns to -2.

[00:03:05]So this is equal to zero. And if you don't mind, you can group this together and then group this. Now we have difference of two cubes and this one.

[00:03:19]But from the difference of two cubes, we have this identity that says that if you have a cube minus b cube, this is a - b and then you multiply a 2 + a b + b 2, right? you multiply this so that if we continue we are going to have a minus b to be x.

[00:03:52]Okay. So from this point we are going to have um a minus b to be x - 2 into a 2 is going to be x² + a b that will be 2 * x and it's 2x + b² that is 2 which is 4 and then we have our x - 2.

[00:04:20]We equate everything to zero. Remember it is the difference of two squares here that give us everything from here down to this point. Now we are going to factoriize what we have because x - 2 is common to both of them. So it comes out here. We have this x 2 + 2x + 4. Right?

[00:04:46]And then remember that um this one divide by itself is one. So here we are going to have + one and everything is still equal to zero.

[00:04:59]So from here now what do you think I am going to do? Add this one first before we proceed. This is x - 2 into x^2 + 2 x + 5 and this is equal to zero. So once you are here the next step to take is to apply your zero product rule because we are multiplying two terms to to get our zero. So you will see that it's either x - 2 is 0 or x^2 + 2x + 5 is = 0.

[00:05:42]So x now is = 0 + 2 or from this part x.

[00:05:50]Okay, we already there. Let's leave that first. So from here now our x is equal to 2 and this is one of the solutions right now to get two more solutions from here we're going to bring it down here um x^2 + 2x + 5 = 0.

[00:06:15]So we're going to use quadratic formula to work on this. So that um okay let me bring the formula - b + or minus we have b^ 2 - 4 a c all over 2 * a.

[00:06:35]Now we need to know our a b c. a is 1, b is 2, c is 5. So we're going to substitute all of them into this. So that x will now be equal to -2 plus or minus b² that is going to be 2^ 2 - 4 * 1 because a is 1 and c is 5.

[00:07:02]Okay. So this is all over 2 * 1 which is the same thing as 2.

[00:07:08]Now our x is -2 + or minus we have 4 - 20 and we are dividing all through by two.

[00:07:21]Okay. So from here x is -2 plus or minus roo<unk> of -6 and this is / 2. Now square root of 16 that we know is four right but the negative here will affect this one. In fact if we want we can remove this negative and then multiply this by<unk> -1.

[00:07:48]So x will now be what - 2 plus or minus roo<unk> of 16 is 4 * root of -1 which is i and this is all over two. So to proceed with what we have x will be -2 + or - 4 i / 2. Okay. So the next point is to divide all through by 2. Yes, two can actually divide and x will be 2 into -2 is -1 plus or minus 2 into 4 i that is going to be 2 i.

[00:08:29]Okay, I think that is it. And this is a twoin one kind of solution. Now let me bring down the equation that we've solved. X^ 4 + X ^ 2 / X = 10. And the solutions we got the first one is that X is equal to 0.

[00:08:52]And this cannot work because if you divide by 0, you're never going to have 10. So this will be rejected. Then the second solution is that x is = 2. This is okay. Then the last two we have x to be = -1 + 2 i. And then x again to be equal to -1 - 2 i. So these are the three solutions that will satisfy.

[00:09:24]accord the first, the second, and the third. Thank you for watching.