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.

 

[blue_raver22]

I would approach this problem by first checking the validity of the given equation. It is important to ensure that the equation is accurate and has been properly derived before attempting to solve it.

Assuming that the equation is valid, I would then proceed to simplify it as much as possible. This may involve getting rid of the roots, as the content suggests, and rearranging the terms. However, it is important to be careful and not make any mistakes during this process.

Once the equation has been simplified, I would then try to find a pattern or relationship between the variables k and l. This can be achieved by manipulating the equation and substituting different values for k and l. This may help in identifying any specific values or ranges for k and l that satisfy the equation.

In this particular case, it may also be helpful to graph the equation and see if there are any intersections with the line \sqrt{k}+\sqrt{l}=\sqrt{4+\sqrt{7}}. This could provide a visual representation of the possible values for k and l.

Overall, solving this equation may require a combination of algebraic manipulation, pattern recognition, and graphical analysis. It may also be helpful to seek assistance from colleagues or consult relevant resources for additional insights and techniques.