To solve this equation, we can start by isolating the variable x.

First, we can divide both sides of the equation by 9 to get rid of the coefficient of (x-1)^2:

(x-1)^2 = 1/4 / 9

Simplifying the right side:

(x-1)^2 = 1/36

Next, we can take the square root of both sides to eliminate the square:

√((x-1)^2) = √(1/36)

Simplifying the right side:

|x-1| = 1/6

Now, we can solve for x by considering two cases:

Case 1: x-1 = 1/6

Adding 1 to both sides:

x = 1/6 + 1

Simplifying:

x = 7/6

Case 2: x-1 = -1/6

Adding 1 to both sides:

x = -1/6 + 1

Simplifying:

x = 5/6

Therefore, the solutions to the equation are x = 7/6 and x = 5/6.