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Math Help - Appoint all the pairs

  1. #1
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    Appoint all the pairs

    Appoint all the pairs (k, l) (both k and l \in R^{+}) such that:
    \sqrt{k}+\sqrt{l}=\sqrt{4+\sqrt{7}}

    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:
    k+l+2\sqrt{kl}=4+\sqrt{7}

    And I don't know what to do next.. Can I please get a little help?
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  2. #2
    Senior Member roninpro's Avatar
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    I would probably rearrange the equation and continue removing the squares.

    2\sqrt{k \ell}-\sqrt{7}=4-k-\ell

    7+4 k \ell-4 \sqrt{7} \sqrt{k \ell}=16-8 k+k^2-8 \ell+2 k \ell+\ell^2

    and so on. Then you'll be left with a quartic equation which you can work with.
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  3. #3
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    Hmm.. thought about it too but when we get to the equation with no roots left at all (I mean, when -4\sqrt{7kl} turns into 112kl), it's REALLY long (and by "REALLY" I mean around 90 characters long). Does it sound right or not really?
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  4. #4
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    I put the (k^{2}+l^{2}-2kl-8k-8l-9)^{2}-112kl=0 to WolframAlpha so that I can see how this function looks like and.. there are no answers. I mean - few of them are saying that l or k shoud be 0 which doesn't fit the requirements of the problem and the others are so extraordinarily long I seriously doubt they may be correct. What am I doing wrong?
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  5. #5
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    Don't you have any ideas, guys? I really can't find any solution and am seriously stuck.
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