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Appoint all the pairs

  1. Sep 12, 2010 #1
    Appoint all the pairs (k, l) (both k and l in R^+) such that:
    [tex]\sqrt{k}+\sqrt{l}=\sqrt{4+\sqrt{7}}[/tex]

    I'm really stuck at it. First of all, I think that getting rid of the roots may be a good idea so we have:
    [tex]k+l+2\sqrt{kl}=4+\sqrt{7}[/tex]
    [tex]2\sqrt{k \ell}-\sqrt{7}=4-k-\ell[/tex]
    [tex]7+4 k \ell-4 \sqrt{7} \sqrt{k \ell}=16-8 k+k^2-8 \ell+2 k \ell+\ell^2[/tex]

    ...but when we get to the equation with no roots left at all (I mean, when [tex]-4\sqrt{7kl}[/tex] turns into 112kl), it's REALLY long (and by "REALLY" I mean around 90 characters long). Does it sound right or not really?
     
  2. jcsd
  3. Sep 12, 2010 #2
    Why can't you just let k be arbitrary, and let l = [√(4+√7) - √k]2, so that √k + √l = √(4+√7) when √(4+√7) - √k > 0?

    The point is that for any fixed k, there is at most one solution for l, since square root is injective.
     
  4. Sep 12, 2010 #3

    Mark44

    Staff: Mentor

    What do you mean by "appoint?" Do you mean, list them?
     
  5. Sep 12, 2010 #4
    I took it to mean: describe all such pairs.
     
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