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The following system of equations comes from a very simple geometric figure I have to deal with a lot at work. Here $r_0,r_1,r_2$ and $L$ are known parameters, and the $x_i$s are the coordinates I'm after.

  1. $x_0^2 + x_3^2 = r_0^2$
  2. $(x_2-L)^2 + x_4^2 = r_2^2$
  3. $x_3 = r_1 + \tfrac{1}{2}x_5(x_0 - x_1)^2$
  4. $x_4 = r_1 + \tfrac{1}{2}x_5(x_2 - x_1)^2$
  5. $x_0 + x_3 x_5(x_0 - x_1) = 0$
  6. $(x_2-L) + x_4 x_5(x_2 - x_1) = 0$

I'm currently solving these numerically. This works perfectly well, but is (a) needlessly expensive if I don't have to do it and (b) extremely unsatisfying given the simple form of my equations.

My question is simply this: is there a piece of open-source (or otherwise free) software I can use to tell me - quickly and for certain - whether this system (and in future possibly others like it) can be solved explicitly in terms of radicals?

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    $\begingroup$ No chance: see the explanation in maplesoft.com/applications/view.aspx?SID=153693 . $\endgroup$
    – user64494
    Oct 6, 2019 at 15:52
  • $\begingroup$ Thanks for this! I'll have a read and play with the demo a bit :) $\endgroup$
    – DCM
    Oct 6, 2019 at 16:03
  • $\begingroup$ The pdf: maplesoft.com/applications/download.aspx?SF=153693/… is really nice. I've heard of Groebner bases but I've never taken the time to learn what they are or what they're used for; nice to have an accessible intro. $\endgroup$
    – DCM
    Oct 6, 2019 at 16:18
  • $\begingroup$ Over the reals? $\endgroup$
    – Rodrigo de Azevedo
    Oct 7, 2019 at 7:57
  • $\begingroup$ Over the reals yes. $\endgroup$
    – DCM
    Oct 7, 2019 at 18:05

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