I am currently solving an algebra and can't figure it out, could anyone help me on this?
$$2\sqrt{N + \sqrt{N^2+4c^2}} = \sqrt{N + \sqrt{N^2+3c^2}} + \sqrt{N + \sqrt{N^2+5c^2}}$$
Which I would like to have a solution to represent N in terms of c, or c in terms of N, either way works. If this is unsolvable, please show me the reason for that.
Thanks in advance.
Of course $c=0$ is a solution. But there are no other real solutions.
After noting that $N=0$ is impossible, we divide both sides by $\sqrt{N}$ and let $c^2/N^2 = t$ to get $$ 2 \sqrt{1+\sqrt{1 + 4 t}} = \sqrt{1+\sqrt{1+3t}} + \sqrt{1+\sqrt{1+5t}} $$ We can write this as $$ 2 g(4 t) = g(3 t) + g(5 t) $$ where $$ g(x) = \sqrt{1+\sqrt{1+x}}$$ Now this function is strictly concave for $x > 0$, as we see by taking its second derivative. Thus since $4 = (3+5)/2$, $g(4t) > (g(3t) + g(5t))/2$ for $t > 0$.