When do we use the quadratic formula, and when the completing the square method?

First, you have to understand that both are essentially the same; the formula is based on the completing the square method. ax^2+b^x+c=0 can with appropriate steps be transformed to x = (-b +- root(b^2-4ac))/2a. The general rule of thumb for when to use each is to check the equation's coefficients. An equation with non-integer coefficients likely means that you have to use the formula. Still, even if the coefficients are all integers, completing the square may not work. Another rule of thumb is to calculate the discriminant (b^2-4ac). If that number is irrational (e.g. root 5), you won't easily solve it with the square completion method. There, you'll have to use the formula. In general, the formula method is more reliable and most of the times quicker than the other.