Using the given equation, take the square root of both sides. Both 169 and 9 are perfect squares, so the left side becomes plus or minus 13/3, which is rational. Six plus 13/3 is a rational number, and 6 minus 13/3 is also a rational number. If the solutions of a quadratic equation are rational, then the equation is factorable.
If you use completing the square and ended up with (x-6)^2, it means it is a perfect square to start off with. That will mean that it can actually be solved by factoring.
