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