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I want to solve following equation:

Solve[{g^2 - gamma*alpha + alpha^2 - 
    omega*(omega*alpha)/(gamma - 2*alpha) - (omega^2*
       alpha^2)/(gamma - 2*alpha)^2 == 0, gamma > 0, omega > 0, 
  g > 0}, alpha]

I want to become an analytic, explicit expression for alpha, but Mathematica gives me only a solution with Root; for example, (there are more solutions):

{alpha -> 
  ConditionalExpression[
   I Root[g^2 gamma^2 - 
        4 g^2 gamma Root[
          g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
               gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
               omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 1] - 
        gamma^3 Root[
          g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
               gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
               omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 1] - 
        gamma omega^2 Root[
          g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
               gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
               omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 1] + 
        4 g^2 Root[
          g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
               gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
               omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 1]^2 + 
        5 gamma^2 Root[
          g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
               gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
               omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 1]^2 + 
        omega^2 Root[
          g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
               gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
               omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 1]^2 - 
        8 gamma Root[
          g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
               gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
               omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 1]^3 + 
        4 Root[g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
               gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
               omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 
          1]^4 + (-4 g^2 - 5 gamma^2 - omega^2 + 
           24 gamma Root[
             g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
                  gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
                  omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 1] - 
           24 Root[
             g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
                  gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + 
                  omega^2) #1^2 - 8 gamma #1^3 + 4 #1^4 &, 
             1]^2) #1^2 + 4 #1^4 &, 2] + 
    Root[g^2 gamma^2 + (-4 g^2 gamma - gamma^3 - 
          gamma omega^2) #1 + (4 g^2 + 5 gamma^2 + omega^2) #1^2 - 
       8 gamma #1^3 + 4 #1^4 &, 1], g > 0 && gamma > 0 && omega > 0]}

This is not an explicit solution. I have also tried Reduce without success.

What can I do?

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  • $\begingroup$ As a response indicates, you can just convert the Root functions into radicals. But actually the Root things themselves are explicit, and for many purposes they are better behaved than parametrized radicals. $\endgroup$ – Daniel Lichtblau Feb 26 '13 at 14:23
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Solve[{g^2 - gamma alpha + alpha^2 - (omega (omega alpha))/(
      gamma - 2 alpha) - (omega^2 alpha^2)/(gamma - 2 alpha)^2 == 0, 
    gamma > 0, omega > 0, g > 0}, alpha, Reals] // ToRadicals // Simplify
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