Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Appoint all the pairs

  1. Sep 12, 2010 #1
    Appoint all the pairs (k, l) (both k and l in R^+) such that:

    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]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


    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook