To solve this equation, we can start by simplifying both sides of the equation.

Expanding the left side:
( x - 6 ) ^ 2 = ( x - 6 ) * ( x - 6 ) = x^2 - 12x + 36

Expanding the right side:
( x - 6 ) ^ 4 = ( x - 6 ) * ( x - 6 ) * ( x - 6 ) * ( x - 6 ) = ( x^2 - 12x + 36 ) * ( x^2 - 12x + 36 )
= x^4 - 24x^3 + 216x^2 - 864x + 1296

Now we have the equation:
x^2 - 12x + 36 = x^4 - 24x^3 + 216x^2 - 864x + 1296

Rearranging the terms and setting the equation equal to zero:
x^4 - 24x^3 + 216x^2 - 864x + 1296 - x^2 + 12x - 36 = 0
x^4 - 24x^3 + 215x^2 - 852x + 1260 = 0

Unfortunately, this equation cannot be easily solved algebraically. It may require numerical methods or approximation techniques to find the solutions.