For the equation to have a solution over R, m>=1 from the constraints placed by the expressions under the radicals.
Rearrange the equation as follows:
sqrt(m - 1/m) = m - sqrt(1 - 1/m)
Square both sides and simplify:
m^2 - m + 1 = 2m*sqrt(1 - 1/m)
Square again and simplify:
m^4 - 2m^3 - m^2 + 2m + 1 = 0
The polynomial factors as:
(m^2 - m - 1)^2 = 0
The two roots of the polynomial are:
(1+sqrt(5))/2 and (1-sqrt(5))/2
Since only one root satisfies the constraint m>=1, there is a unique solution of m=(1+sqrt(5))/2
The unique solution
