How can we determine stationary points by completing the square?

Suppose we have completed the square on y=ax^2+bx+c and attained y=a(x+p)^2+q, where a,b,c,p,q are real numbers with 'a' not equal to zero and p,q can be expressed in terms of a,b,c. For a>0 we have a minimum point, where x takes a value such that a(x+p)^2+q is smallest, giving x= -p and hence y=q. For a<0, we have a maximum point, where x takes a value such that a(x+p)^2+q is biggest, also giving x= -p and hence y=q.