A Very Quartic Equation | Can You Solve?

[@SyberMath]

Proving a quartic inequality (Sum of Fourth Powers): kzbin.info/www/bejne/pHTFpqWgqZaInbc

[@davidseed2939]

4:50 if z=1, then xy=1 gives x=y= 1 OR x=y=-1. ie a fourth solution. This corresponds with the fact that the equation is symmetric in x,y,z and so the solution must be symmetric. ie the solution set is x=y=z=1 and all permutations of (x,y,z)=(1,1,-1)

[@snejpu2508]

I used AM>=GM formula: (x^4 + y^4 + z^4 + 1)/4 = 4throot(x^4 * y^4 * z^4 * 1). It works, as x^4, y^4 and z^4 are always nonnegative. After a few steps, you get x^4 + y^4 + z^4 >= 4xyz - 1. So, the relation becomes an equation, if x^4 = y^4 =z^4. It is still a little tricky, as you have to be careful with the signs. It's not the same as x=y=z apparently, as it only gives the solution (1,1,1)... I'm not sure how you could find the rest of solutions in an universal way from there. Or you could just try out all 3 possibilities with +/- 1 and see, what happens.

[@SyberMath]

Good to see you! ☺️

[@snejpu2508]

@@SyberMath Hi! I often follow your content, but only comment on problems not too obvious and not too difficult for me. ;-)

[@ayushrudra8600]

4:50 you missed (-1, -1, 1)

[@SyberMath]

😄😄

[@mr.d8747]

*Yeah, a quartic is supposed to have four solutions*

[@Hobbitangle]

The equation is symmteric over x,y,z So if there is a solution {1,-1,-1} there should be the solutions {-1,1,-1} and {-1,-1,1}. You did miss one. And definitely the complex solutions should be considered as well.

[@Hobbitangle]

Complex solutions examples: {-1,i,i},{1,i,-i} because i⁴=1 -1•i•i=1 1•i•(-i)=1

[@SyberMath]

😄😄

[@NadiehFan]

I wonder how at 4:48 you could have missed the solution (−1, −1, 1) of your equation in x, y, z when at the end you show the result from WA which clearly finds _four_ integer solutions.

[@SyberMath]

😮😄 that's me!

[@tixanthrope]

arithmetic mean >= geometric mean yields (1, 1, 1) other solutions are (1, -1, -1) etc.

[@jurem2978]

Me seeing the problem, be like: "it's AM GM, it's AM GM"

[@ayushrudra8600]

Bro why are some people hating so hard in the comments when he accidentally missed a solution

[@NadiehFan]

I found two videos where you already did both problems from this video, but KZbin won't let me post the links because if I do my message gets removed immediately. But anyway, here are the video IDs: ogfNGLyL7NA pFUVGjMUHMY

[@SyberMath]

Thanks for the heads up! 😍 One of them is a generalization or a version with four variables. The other one is the same problem done slightly differently, maybe! 😜 Here's the links: kzbin.info/www/bejne/pZjJf3qCrrFqhKM kzbin.info/www/bejne/pne4h3qggrp7g7s

[@MochooCheung-pu8js]

Giving this kind of question is Cheating, because the question maker already knows the answer.

[@SyberMath]

ehehehehehe 😄😜