First use the distributive property to write x(6x-5)=6 as 6x^2-5x=6. Next factor the coefficient of x^2 out on the left side to write it as 6(x^2-(5/6)x)=6.
We can now complete the square: Take the coefficient of x inside the parentheses, the -5/6, divide it by 2 to get -5/12, then square that number to get 25/144.
Next, add the 25/144 inside the parentheses on the left and compensate for that (balance the equation) by adding 6*25/144=25/24 to the right hand side:
6(x^2-(5/6)x+25/144)=6+25/24=(144+25)/24=169/24.
The reason this is a good idea is that this technique has made the expression inside the parentheses on the left a perfect square. This last equation can be written as
6(x-5/12)^2=169/24 (note also the appearance of the -5/12 again...this is no coincidence).
To finish, divide both sides by 6 to get (x-5/12)^2=169/144 and then take the square root of both sides, allowing a \pm sign on the right to get x-5/12=\pm 13/12.
Now add 5/12 to everything to get x=5/12 \pm 13/12. The two solutions are x=18/12=3/2 and x=-8/12=-2/3.
