Completing the square is covered in post 3 of this topic: http://www.mathhelpforum.com/math-he...tic-curve.html
Comparing coefficients says that two equations are equal if and only if the coefficients on each side match.
For example given that ax^2+bx+c = x^2 - 12x+1 comparing coefficients tells you that a = 1, b=-12 and c = 1