Completing the square
Contents
Motivations
Quadratic equations are usally hard to solve, if they do not factor.
Some of them, in forms 
, are easy to solve by taking the squareroot of b and subracting a. Completing the square is a technique to munipulate every quadratic into the easily solve-able form above.
General Solution For A Monic Quadratic
Let the quadratic be in the form 
.
Moving be to the other side, we obtain
Then, adding a^2/4 to each side and factoring, we get
Which is solvable as described above.
Applications of Adding and Factoring
Other degrees of polynomials may be solved by adding constant terms and factoring.
Another common usage is in conic sections. The equations for conic sections typically contain a squared term such as 
. However, the problem may be posed as to convert from an expanded form to a factored perfect square. Completing the square is the standard method.
All kinds of exotic factoring techniques are used on the AIME, and completing the square is just one. The follow solution to AIME 1987 #5 is but one.
Instructive Problem: AIME 1987 #5
Problem:m, n are integers such that 
. Find 
.
Outline Solution: We use the techinique of adding to factor.
Rearrange to 
. The key step is changing the equation to 
, where the equation factors to 
, from which the problem is trivial to solve by applying some simple number theory.
