completing the square

Completing the square is a way to simplify polynomial expressions by writing them as sums or differences of squares. The method goes like this:

1. Starting with an expression such as this:

   Ax² + Bx + C

2. Divide through by A so that the x² term has a coefficent of 1.

   x² + (B / A)x + (C / A)

3. Add and subtract one half (B / A) squared.

   x² + (B / A)x + (B / (2A))² + (C / A) - (B / (2A))²

4. The first three terms of the above expression can be written as

   (x + (B / (2A)))²

   This can be verified by multiplying the expression out.
5. The finished formula can be written as

   (x + (B / (2A)))² + (C / A) - (B / (2A))²

6. Multiplying out the last two terms gives us a total result of:

   (x + (B / (2A)))² + (4AC - B²) / (4A²)