NMinimize[{Abs[
Sqrt[(((x Sqrt[3 - 4 x^2])/Sqrt[1 + 4 x^2] +
Sqrt[(8 x^2 + Sqrt[3 + 8 x^2 - 16 x^4])/(2 + 8 x^2)] -
1/2 Sqrt[(7 + 8 Sqrt[3 - 4 x (1 + x)] -
8 x (-1 + x (1 + 2 x (2 + x))))/(13 +
8 x (1 + x) (-3 + 2 x (1 + x)))])^2 + (-(1/2) -
1/2 (-1 +
Sqrt[3 - 4 x (1 + x)]) Sqrt[(7 +
8 Sqrt[3 - 4 x (1 + x)] -
8 x (-1 + x (1 + 2 x (2 + x))))/(13 +
8 x (1 + x) (-3 + 2 x (1 + x)))])^2)] - 1], .1 <
x < .3}, x, AccuracyGoal -> 2000, WorkingPrecision -> 4000]
It was right in the notebook. The big Sqrt[] doesn't copy over correct as input text.