To solve this equation, we first need to expand the left side of the equation:

(x - 1/6)^2 = (x - 1/6)(x - 1/6)
= x^2 - x/6 - x/6 + 1/36
= x^2 - x/3 + 1/36

Now we can set this equal to 49/81:

x^2 - x/3 + 1/36 = 49/81

To get rid of the fractions, we can multiply both sides of the equation by 36:

36(x^2 - x/3 + 1/36) = 36(49/81)
36x^2 - 12x + 1 = 196/3
36x^2 - 12x + 1 = 65.3333

Now we have a quadratic equation that we can solve using the quadratic formula:

x = (-(-12) ± √((-12)^2 - 4(36)(1)))/(2(36))
x = (12 ± √(144 - 144))/72
x = (12 ± 0)/72
x = 12/72
x = 1/6

Therefore, the solution to the equation is x = 1/6.