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Algebra with a complicated function

  1. Jun 3, 2008 #1
    1. The problem statement, all variables and given/known data

    [tex]
    \frac{\pi(1+2\alpha)}{t}=x \&\& ((\alpha \geq 0 \&\& 1+ 2\alpha < 4\beta \&\& \pi \sqrt{-(1+2\alpha)^2+16\beta}=2t)\\ ||((1+2\alpha>0 \&\& 2\alpha < 1 +4 \beta \&\& \beta \geq 0 \&\& \pi \sqrt{(-1+2\alpha -4\beta) (3+2 \alpha +4\beta)}=2t)
    [/tex]

    [tex]
    \alpha and \beta[/tex]are integers.

    This is a solution I obtained from Mathematica, it's ugly as you can tell. How can I generalize this concisely?
    How can I find a general expression for x from this equation?

    2. Relevant equations



    3. The attempt at a solution

    If we look at the first condition, [tex] \alpha \geq 0 [/tex] which is nice
    \alpha \beta 1+2*alpha < 4*beta
    0 1 1 < 4
    1 2 3 < 8
    2 3 5 < 12


    This is false when
    \alpha \beta 1+2*alpha < 4*beta
    0 0 1 < FALSE
    1 1 3 < 4
    2 2 5 < 8

    Any suggestions?
     
    Last edited by a moderator: Aug 17, 2014
  2. jcsd
  3. Jun 3, 2008 #2
    Based on the equality for condition 1, we also know that [tex]\beta >0[/tex] but how do you generalize the order?

    Can this even be generalized to start with? I think not, but how can you tell? There are two condition I have to take into account.
     
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