Expanding the left side of the equation:

(x-6)(x+6) = x^2 + 6x - 6x - 36 = x^2 - 36

(2x-3)(x-1) = 2x^2 - 2x - 3x + 3 = 2x^2 - 5x + 3

Therefore,

(x-6)(x+6) - (2x-3)(x-1) = x^2 - 36 - (2x^2 - 5x + 3) = x^2 - 36 - 2x^2 + 5x - 3 = -x^2 + 5x - 39

Setting this equal to the right side of the equation, which is 6-x^2:

-x^2 + 5x - 39 = 6 - x^2
5x - 39 = 6
5x = 45
x = 9

Therefore, the value of x that satisfies the equation is x = 9.