Maximizing Pairs (k,l) in R^+: k+l+2\sqrt{kl}=4+\sqrt{7}

[Foamy]

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:
$$k+l+2\sqrt{kl}=4+\sqrt{7}$$
$$2\sqrt{k \ell}-\sqrt{7}=4-k-\ell$$
$$7+4 k \ell-4 \sqrt{7} \sqrt{k \ell}=16-8 k+k^2-8 \ell+2 k \ell+\ell^2$$

...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?

 

[adriank]

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.

 

[Mark44]

What do you mean by "appoint?" Do you mean, list them?

 

[adriank]

I took it to mean: describe all such pairs.