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Solve[beta*x *((1 + (1 - m)*sigma*theta ((x^(-1/2) - 1)/((1 - theta)*x^(-1/2) + theta)))) - 
   S == 0, x]

Can someone please help me, I don't know why x is not solved . Thank you so so much!!

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closed as off-topic by zhk, MarcoB, Roman, garej, Henrik Schumacher Jun 7 at 11:26

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – zhk, Roman, Henrik Schumacher
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ Use Solve with a capital S $\endgroup$ – zhk Jun 7 at 4:08
  • $\begingroup$ Thank you zhk, I fixed the Solve, but still have odd long long results... $\endgroup$ – Diana Yang Jun 7 at 4:11
  • $\begingroup$ LeafCount[Solve[...]]==35541 but LeafCount[Simplify[Solve[...]]]==2054 so that makes the solution 17 times "simpler." FullSimplify might do even more, but it will be very very slow to finish that. $\endgroup$ – Bill Jun 7 at 5:27
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The three solutions are the squares of the three roots of a third-order polynomial:

sol = {Root[S*(θ-1)-S*θ*#+β*(1-θ+θ*σ-m*θ*σ)*#^2+β*θ*(1-σ+m*σ)*#^3 &, 1]^2,
       Root[S*(θ-1)-S*θ*#+β*(1-θ+θ*σ-m*θ*σ)*#^2+β*θ*(1-σ+m*σ)*#^3 &, 2]^2,
       Root[S*(θ-1)-S*θ*#+β*(1-θ+θ*σ-m*θ*σ)*#^2+β*θ*(1-σ+m*σ)*#^3 &, 3]^2}

I don't think the solutions can be written any more succintly. To convert them into explicit formulas (Cardano formulas), use the ToRadicals command.

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