Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Ishaan94

  • 3 years ago

Find all real triples \((x,y, z)\) that satisfy \(x^4+y^4+z^4−4xyz = −1\).

  • This Question is Closed
  1. Ishaan94
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @KingGeorge

  2. mahmit2012
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It is just a surface in a 3 dimensional space.

  3. KingGeorge
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I've seen a method to solve problems like this easily, but I can't remember it right now. I'll try and look it up later to try and find more information.

  4. mahmit2012
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1341442265927:dw|

  5. mahmit2012
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1341442357328:dw|

  6. mahmit2012
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1341442492121:dw|

  7. mahmit2012
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So I just found out min(f)=-1 so as I told in above that is a surface in a space.

  8. mukushla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hi @Ishaan94 Using Completing the Square \[x^4+y^4+z^4-4xyz+1=0\\x^4-2x^2y^2+y^4+2x^2y^2+z^4+2z^2-2z^2-4xyz+1=0\\x^4-2x^2y^2+y^4+2(x^2y^2-2xyz+z^2)+z^4-2z^2+1=0\\(x^2-y^2)^2+2(xy-z)^2+(z^2-1)^2=0\] so \[x^2-y^2=xy-z=z^2-1=0\]

  9. mukushla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    only triples that satisfies the equation \[(x,y,z)=(1,1,1),(−1,−1,1),(1,−1,−1),(−1,1,−1)\]

  10. Ishaan94
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Thanks @mukushla if it's not too much to ask did you take training or something for the olympiads?

  11. mukushla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    welcome my friend no im a chemical engineering student and just love math

  12. mukushla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    do u wanna get ready for Math Olympiad or something?

  13. Ishaan94
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no i am already past my high school but i love solving math problems.

  14. mukushla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    just like me :D

  15. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy