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I am trying to solve a more general question and I have the following subproblem: Find $x>0$ that satisfies for fixed $ i \geq 3$, $$\left(1 + \frac{1}{b^2}\right) x = \frac{\sum_i a_i^2} {b^2} + \frac{1}{b} \sum_i \sqrt{-a_i^2 + x + b^2x}.$$

I have tried commercial solvers (Mathematica) but they can't find a solution (they stop running after an hour or so). I have tried several transformations, for example using the identity $\sqrt{p} + \sqrt{q} = \sqrt{p+q + \sqrt{4pq}}$ or taking squares of both sides, but the equation becomes more intractable.

Do you have any pointers or ideas?

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    $\begingroup$ For example, the solution to $x = \sqrt{3 + x} + \sqrt{2 + x} + \sqrt{1+x}$ is an algebraic integer of degree 8 (the unique positive solution is about 10.67), and I suspect that for sufficiently random integer constants your solution will be an algebraic number of degree $2^i$. It doesn't seem reasonable to solve the equation exactly. Are you having difficulty finding a numerical solution? $\endgroup$
    – Jack Huizenga
    Commented May 28 at 19:13
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    $\begingroup$ Are you sure this is the equation you are trying to solve? Note that when the $a_i$'s are non-zero and $x\rightarrow 0$, the right hand side is complex. $\endgroup$
    – kindasorta
    Commented May 28 at 19:13
  • $\begingroup$ It's a step of a more general problem I'm trying to solve. I have done a lot of work to reduce the problem to this single equation. I need an explicit solution. The original problem is here: mathoverflow.net/questions/471075/… $\endgroup$
    – Margot.
    Commented May 28 at 19:29

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