Here's one way, the "ac" method.
x^2 - 13x + 36 = 0
Multiply the coefficient of x^2 by the constant term: 1*36 = 36
Now write down all the ways to factor 36:
36 = 1*36, -1*-36
36 = 2*18, -2*-18
etc.
Now look at your list and find the two factors that add up to the coefficient of the x term -13:
36 = -4*-9; -4 + -9 = -13
So this is what we do:
x^2 - 13x + 36 = 0
x^2 + (-4x + -9x) + 36 = 0 <--- We could have used (-9x + -4x) also.
(x^2 - 4x) + (-9x + 36) = 0 <--- This part is called "factoring by grouping"
x(x - 4) + -9(x - 4) = 0
Note that each term has a factor of x - 4 in it, so factor an x - 4 out on the right:
(x - 9)(x - 4) = 0
Thus either x - 9 = 0 or x - 4 = 0. Thus x = 4 or x = 9.
This method will always work unless the quadratic cannot be factored over the rational numbers, that is to say that it won't work if the solutions are (non-rational) real or complex numbers.
-Dan