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how many solutions?

Discussion in 'Mathematics' started by coffee-breff, Mar 29, 2012.

  1. coffee-breff

    coffee-breff New commenter

    I was doing some work with an very able year 10 class and set this for them
    (x^2+5x+5)^(x^2+11x+30)=1
    what are the solutions?
     
  2. How did they get on?
     
  3. coffee-breff

    coffee-breff New commenter

    they found 4, some with algebra and some by graphing it. I have left them to find the others over Easter. They are convinced that there are no more.
     
  4. Scifihel

    Scifihel New commenter

    I get four solutions.


    hang on...
    just as well we can edit posts

    thinking again, with pen and paper
     
  5. Karvol

    Karvol Occasional commenter

    Are you posing it as a question for the forum or are you posing it because you need to find out?
    More importantly, what did your class learn from it?

     
  6. Scifihel

    Scifihel New commenter

    Formatting nightmare, with stupid keyboard.
     
  7. Karvol

    Karvol Occasional commenter

    -6, -5, -4 and -1 by observation. What others are there ( I need to go and cook dinner... )?
     
  8. coffee-breff

    coffee-breff New commenter

    i think I know the answer, I just thought it was an interesting problem.
     
  9. Karvol

    Karvol Occasional commenter

    And -3 and -2 because the "base" can also be -1 if the "power" is even.

     
  10. Karvol

    Karvol Occasional commenter

    It looks like my Year 10 class have just got their first homework for after the holidays. Should make for some interesting class discussion...
    Thanks!
     
  11. They were mine. Had to think about the -1.
    I would expect most half OK year 12/13s to do it. Year 10s? depends on their approach.....
     
  12. Are there not infinitely many? i.e. all the solutions that correspond to:
    u^0 = 1
    1^n=1
    (-1)^(2n)=1
    with some restrictions if you want real solns? (Nice question, BTW)
     
  13. Don't forget, u and n are not independent. If we force the exponent to be 0, then this fixes u.
    By taking logs, I got the 6 solutions stated. Drawing a graph seems to confirm this.
     
  14. Yes, you're right. That'll teach me not to solve problems in my head, and to show all of my working.
     
  15. Looked at this with further maths today, we came to the conclusion that the theoretical max no. of solns is 10. 2 from the power and 8 from putting base = 1, -1, i, -i provided the power then met certain criteria. Do you agree?
     
  16. What were your solutions for x when the base was i, and -i?
     


  17. x= -1/2 (√113+11),
    x= 1/2 (√113-11),
    x= -1/2 (√129+11),
    x= 1/2 (&radic;129-11)<font size="3" face="Times New Roman">

    </font>
    ?
     
  18. I think a 'proof' that there are only 6 solutions might be along the lines of:
    * We have a solution if and only if the base is the nth root of unity and the power is a multiple of n.
    * However, any root of unity not 1 or -1 or i or -i will lead to an non integer value in the power because of the 11x term. Thus n must be 4, or we take the integer roots when n is a multiple of 4.
     

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