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I tried to run the following code to get the region where (v,b,r) satisfies some complicated set of inequalities.

Resolve[Exists[{d}, (v + b (1 - r) (d)^0.5 - 
    d - ((v + b (1 - r) (d)^0.5 - d)^2 - v^2)^0.5) (v + 
    b (1 + r) (d)^0.5 - 
    d + ((v + b (1 - r) (d)^0.5 - d)^2 - v^2)^0.5) - 
 4 (v/2 + r^2 b^2/8)^2 == 0 && d > 0 && d < v&& ((3/4 b^2 r^2 < v < 3/4 b^2 && 
   r < (8 v*b^2 + b^4)/(4 v + b^4)) || (v > 3/4 b^2 && 
   r < ((1 + b^2) v + b^4/4)/(4 v + b^4))) && 0 < b < 1 && v > 0 &&
r > 0],Reals]

However, Mathematica always return the set of existence conditions without giving me the numerical approximation. There is no sign of any evaluation.

Can anyone tell me why Resolve fails to work in this situation?

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As the help page states, Resolve[] "attempts to resolve expr into a form that eliminates ForAll and Exists quantifiers." If it is not successful, it returns with the quantifiers still in there.

Regarding why it might not have succeeded, please note that you are making the problem finite precision by using 0.5 . I have changed these to (1/2), and Mathematica is still thinking about it ... . (With the (1/2), you might try an easier form of the problem. BOL!

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