Given the statement √4x^2-4x+1 = 2x-1

 √ [4x^2  - 4x + 1 ]  can be wriiten as  √ [ 2x - 1 ]^2   =   l  2x - 1 l

So we have that either

2x - 1  = 2x - 1

Redundant

Or

2x - 1 = - [  2x - 1 ]

2x - 1 == -2x + 1      rearrange  as

4x  - 2  = 0

4x  = 2

x  = 2/4  = 1/2

The root will be  ≥ 0    for all x   and at x  = 1/2 the root   = 0

But   at x  = 1/2....2x -1   = 0       and this function  wll be negative  for all x < 1/2

So...the 4 largest integer values for 2x - 1  for which the equality is not true are     -3, -2, -1 , 0

And the average of these is  -2