x + 1/x = 5 multiply through by x
x^2 + 1 = 5x
x^2 - 5x = -1 complete the square on x
x^2 - 5x + 25/4 = -1 + 25/4
(x - 5/2)^2 = 21/4 take the positive root [ we will get the same result with the negative root ]
x - 5/2 = √21/2
x = [ √21 + 5 ] / 2
So ( x - 2) = ( √21 + 1 ] / 2
And ( x - 2)^2 = [ 21 + 2√21 + 1 ] / 4 = [ 22 + 2√21] / 4 = [ 11 + √21] / 2
So
(x - 2)^2 + 25 /( x - 2)^2 =
[ 11 + √21] / 2 + 50/ [ 11 + √21] =
[rationalize the denominator of the second fraction by multiplying top/bottom by 11 - √21 ]
[ 11 + √21 ] / 2 + 50 [ 11 - √21] / [ 121 - 21] =
[ 11 + √21 ] / 2 + 50 [11 - √21] / 100 =
[ 11 + √21 ] / 2 + [11 - √21 ] / 2 =
[11 + 11 ] / 2 =
22 / 2 =
11