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I am trying to solve this equation where I need the solution of K in term of v

 Solve[1 - 
   K - (54 (20 - 
        K) v (2 (-10 + 
          K) (5 (1300 - 10 K + 3 K^2 - 100 (2 + K)) Hypergeometric2F1[
            3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
            3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
            1, (-10 + K)/(-20 + K)] + (-13000 + 400 K - 75 K^2 + 
             6 K^3 + 400 (-25 + 7 K) - 
             10 (200 - 140 K + 21 K^2)) Hypergeometric2F1[
            3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
            3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
            2, (-10 + K)/(-20 + K)]) (((-20 + 
            K)^3 Hypergeometric2F1[-1 - Sqrt[3], -1 + Sqrt[3], 
           1, -(10/(-20 + K))])/(-10 + K)^3)))/((12 (-20 + K) (-10 + 
          K) (5 (1300 - 10 K + 3 K^2 - 100 (2 + K)) Hypergeometric2F1[
            3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
            3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
            1, (-10 + K)/(-20 + K)] + (-13000 + 400 K - 75 K^2 + 
             6 K^3 + 400 (-25 + 7 K) - 
             10 (200 - 140 K + 21 K^2)) Hypergeometric2F1[
            3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
            3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
            2, (-10 + K)/(-20 + K)]) (-(((20 - 
             K)^3 Hypergeometric2F1[-1 - Sqrt[3], -1 + Sqrt[3], 
            1, -(K/(-20 + K))])/(8 (-10 + K)^3))) + 
       1/4 (-3 (-20 + K) (-10 Sqrt[3] - Sqrt[(-10 + K)^2] + 
             Sqrt[3] K) (-10 Sqrt[3] + Sqrt[(-10 + K)^2] + 
             Sqrt[3] K) (-K (2 K - 2 (10 + K)) Hypergeometric2F1[
               2 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
               2 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 1, (
               2 (-10 + K))/(-20 + K)] - 
             2 (K^2 - 2 K (10 + K) + 
                4 (100 - 10 K + K^2)) Hypergeometric2F1[
               2 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
               2 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 2, (
               2 (-10 + K))/(-20 + K)]) - 
          2 (-10 + 
             K)^2 (-K (3 K^2 - 2 K (50 + K) + 
                4 (300 - 10 K + K^2)) Hypergeometric2F1[
               3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
               3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 1, (
               2 (-10 + K))/(-20 + K)] - (10 (-28 + K) K^2 + 3 K^3 + 
                8 K (1050 - 70 K + K^2) + 
                16 (-6000 + 750 K - 40 K^2 + K^3)) Hypergeometric2F1[
               3 - (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 
               3 + (Sqrt[3] (-10 + K))/Sqrt[(-10 + K)^2], 2, (
               2 (-10 + K))/(-20 + K)])) )) == 0, K]

But, everytime I am getting a statement Solve::nsmet: This system cannot be solved with the methods available to Solve.

Please give me any suggestion. In have tried to find solution by using 'FindRoot'. But the problem with this is, If I give some value of 'v' then only it gives root.

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Massages from Solve says: transcendental equation can't be solved analytically,but we can plot solution of function k[v].

eq = 1 - k - (108 (20 - k) (-20 + k)^3 v Hypergeometric2F1[-1 - Sqrt[
  3], -1 + Sqrt[3], 
 1, -(10/(-20 + 
   k))] (5 (1300 - 10 k + 3 k^2 - 100 (2 + k)) Hypergeometric2F1[
    3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
    3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
    1, (-10 + k)/(-20 + k)] + (-13000 + 400 k - 75 k^2 + 6 k^3 + 
     400 (-25 + 7 k) - 
     10 (200 - 140 k + 21 k^2)) Hypergeometric2F1[
    3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
    3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
    2, (-10 + k)/(-20 + k)]))/((-10 + 
  k)^2 (-(1/(2 (-10 + k)^2))
   3 (20 - k)^3 (-20 + k) Hypergeometric2F1[-1 - Sqrt[3], -1 + 
      Sqrt[3], 
     1, -(k/(-20 + 
       k))] (5 (1300 - 10 k + 3 k^2 - 
         100 (2 + k)) Hypergeometric2F1[
        3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
        3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
        1, (-10 + k)/(-20 + k)] + (-13000 + 400 k - 75 k^2 + 
         6 k^3 + 400 (-25 + 7 k) - 
         10 (200 - 140 k + 21 k^2)) Hypergeometric2F1[
        3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
        3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
        2, (-10 + k)/(-20 + k)]) + 
  1/4 (-3 (-20 + k) (-10 Sqrt[3] - Sqrt[(-10 + k)^2] + 
        Sqrt[3] k) (-10 Sqrt[3] + Sqrt[(-10 + k)^2] + 
        Sqrt[3] k) (-k (2 k - 2 (10 + k)) Hypergeometric2F1[
          2 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
          2 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 1, (
          2 (-10 + k))/(-20 + k)] - 
        2 (k^2 - 2 k (10 + k) + 
           4 (100 - 10 k + k^2)) Hypergeometric2F1[
          2 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
          2 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 2, (
          2 (-10 + k))/(-20 + k)]) - 
     2 (-10 + 
        k)^2 (-k (3 k^2 - 2 k (50 + k) + 
           4 (300 - 10 k + k^2)) Hypergeometric2F1[
          3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
          3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 1, (
          2 (-10 + k))/(-20 + k)] - (10 (-28 + k) k^2 + 3 k^3 + 
           8 k (1050 - 70 k + k^2) + 
           16 (-6000 + 750 k - 40 k^2 + k^3)) Hypergeometric2F1[
          3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
          3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 2, (
          2 (-10 + k))/(-20 + k)]))));
ContourPlot[eq == 0, {v, -10, 1}, {k, 0, 11}, FrameLabel -> Automatic]

enter image description here

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Addendum to the answer @MariuszIwaniuk. It is possible to construct a function for calculating $k(v)$ with a given WorkingPrecision

eq = 1 - k - (108 (20 - k) (-20 + k)^3 v Hypergeometric2F1[-1 - 
        Sqrt[3], -1 + Sqrt[3], 
       1, -(10/(-20 + k))] (5 (1300 - 10 k + 3 k^2 - 
           100 (2 + k)) Hypergeometric2F1[
          3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
          3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
          1, (-10 + k)/(-20 + k)] + (-13000 + 400 k - 75 k^2 + 
           6 k^3 + 400 (-25 + 7 k) - 
           10 (200 - 140 k + 21 k^2)) Hypergeometric2F1[
          3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
          3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
          2, (-10 + k)/(-20 + k)]))/((-10 + 
         k)^2 (-(1/(2 (-10 + k)^2)) 3 (20 - k)^3 (-20 + 
           k) Hypergeometric2F1[-1 - Sqrt[3], -1 + Sqrt[3], 
          1, -(k/(-20 + k))] (5 (1300 - 10 k + 3 k^2 - 
              100 (2 + k)) Hypergeometric2F1[
             3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
             3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
             1, (-10 + k)/(-20 + k)] + (-13000 + 400 k - 75 k^2 + 
              6 k^3 + 400 (-25 + 7 k) - 
              10 (200 - 140 k + 21 k^2)) Hypergeometric2F1[
             3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
             3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
             2, (-10 + k)/(-20 + k)]) + 
        1/4 (-3 (-20 + k) (-10 Sqrt[3] - Sqrt[(-10 + k)^2] + 
              Sqrt[3] k) (-10 Sqrt[3] + Sqrt[(-10 + k)^2] + 
              Sqrt[3] k) (-k (2 k - 2 (10 + k)) Hypergeometric2F1[
                2 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
                2 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
                1, (2 (-10 + k))/(-20 + k)] - 
              2 (k^2 - 2 k (10 + k) + 
                 4 (100 - 10 k + k^2)) Hypergeometric2F1[
                2 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
                2 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
                2, (2 (-10 + k))/(-20 + k)]) - 
           2 (-10 + 
               k)^2 (-k (3 k^2 - 2 k (50 + k) + 
                 4 (300 - 10 k + k^2)) Hypergeometric2F1[
                3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
                3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
                1, (2 (-10 + k))/(-20 + k)] - (10 (-28 + k) k^2 + 
                 3 k^3 + 8 k (1050 - 70 k + k^2) + 
                 16 (-6000 + 750 k - 40 k^2 + k^3)) Hypergeometric2F1[
                3 - (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
                3 + (Sqrt[3] (-10 + k))/Sqrt[(-10 + k)^2], 
                2, (2 (-10 + k))/(-20 + k)]))));


f[x_, p_] := 
 Block[{v = x, $MinPrecision = p, $MaxPrecision = p}, 
  Chop[k /. FindRoot[eq == 0, {k, 1}, WorkingPrecision -> p]]]

Then, for example, we have

f[-5, 20]

(*Out[]= 0.040304506890271919987*)

We can combine two solutions

lst = Table[{x, f[x, 30]}, {x, -10, 0, 1}];

fig1 = ListPlot[lst, PlotStyle -> Red];
fig2 = ContourPlot[eq == 0, {v, -10, 0}, {k, 0, 1}, 
  FrameLabel -> Automatic, PlotPoints -> 50];
Show[fig2, fig1]

Figure 1

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