To solve the equation, we can start by expanding the expression:

(x-1)^2 - x(4-x) = 0

Expanding the square:

(x^2 - 2x + 1) - x(4-x) = 0

Distributing the x:

x^2 - 2x + 1 - 4x + x^2 = 0

Combining like terms:

2x^2 - 6x + 1 = 0

Now we can solve this quadratic equation. We can either factor it or use the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 2, b = -6, and c = 1. Plugging these values into the formula:

x = (-(-6) ± √((-6)^2 - 4(2)(1))) / (2(2))

Simplifying:

x = (6 ± √(36 - 8)) / 4

x = (6 ± √28) / 4

x = (6 ± 2√7) / 4

Simplifying further:

x = (3 ± √7) / 2

So the solutions to the equation are:

x = (3 + √7) / 2

x = (3 - √7) / 2