To simplify the expression E=√(1-4x+4x^2)-2x, we can start by expanding the square root term.

E = √(1-4x+4x^2) - 2x
E = √((1-2x)^2) - 2x
E = |1-2x| - 2x

Now, we can consider two cases:

Case 1: 1-2x ≥ 0
In this case, the absolute value becomes unnecessary.

E = (1-2x) - 2x
E = 1 - 4x

Case 2: 1-2x < 0
In this case, the absolute value becomes negative.

E = -(1-2x) - 2x
E = -1 + 2x - 2x
E = -1

Therefore, the simplified expression for E=√(1-4x+4x^2)-2x is:

E = 1 - 4x, when 1-2x ≥ 0
E = -1, when 1-2x < 0