First, you can get each fraction over a common denominator, in this case #x(x - 2)#, so the fractions can be added:
#((x-2)/(x-2))(9/x) + (x/x)(9/(x-2)) = 12#
#(9(x-2) + 9x)/(x(x-2)) = 12#
We can now expand and add the numerator:
#(9x-18 + 9x)/(x(x-2)) = 12#
#(18x-18)/(x(x-2)) = 12#
We can now eliminate the fraction through multiplication on each side of the equation to keep the equation balanced:
#(18x-18)/(x^2-2x) = 12#
#(x^2 - 2x)(18x - 18)/(x^2-2x) = 12(x^2 - 2x)#
#cancel((x^2 - 2x))(18x - 18)/cancel((x^2-2x)) = 12x^2 - 24x#
#18x - 18 = 12x^2 - 24x#
We can now make a single quadratic and factor:
#18x - 18 - 18x + 18 = 12x^2 - 24x - 18x + 18#
#12x^2 - 42x + 18 = 0#
#(6x - 3)(2x - 6) = 0#
We can now solve each factor for #0#:
#6x - 3 = 0#
#6x - 3 + 3 = 0 + 3#
#6x = 3#
#(6x)/6 = 3/6#
#x = 1/2#
and
#2x - 6 = 0#
#2x - 6 + 6 = 0 + 6#
#2x = 6#
#(2x)/2 = 6/2#
#x = 3#
