# Thread: Appoint all the pairs

1. ## 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?

2. 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.

3. 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?

4. 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?

5. Don't you have any ideas, guys? I really can't find any solution and am seriously stuck.