ufuncs = Solve[1+u*x-2*u^2==(1-x)*Sqrt[1-u^2],u]

uofx1[x_] = u /. ufuncs[[1]]

Plot[uofx1[x],{x,0,1}]

uofx2[x_] = u /. ufuncs[[2]]

Plot[uofx2[x],{x,0,1}]

uofx3[x_] = u /. ufuncs[[3]]

Plot[uofx3[x],{x,0,1}]

uofx4[x_] = u /. ufuncs[[4]]

Plot[uofx4[x],{x,0,1}]

rra[x_] := -x*uofx3''[x]/uofx3'[x]

Plot[rra[x],{x,0,1}]

rra[0.0001]

rra[0.125]

rra[0.9999]