completing the square
Let us consider the expression ${x}^{2}+xy$, where $x$ and $y$ are real (or complex) numbers. Using the formula^{}
$${(x+y)}^{2}={x}^{2}+2xy+{y}^{2}$$ 
we can write
${x}^{2}+xy$  $=$  ${x}^{2}+xy+0$  
$=$  ${x}^{2}+xy+{\displaystyle \frac{{y}^{2}}{4}}{\displaystyle \frac{{y}^{2}}{4}}$  
$=$  ${\left(x+{\displaystyle \frac{y}{2}}\right)}^{2}{\displaystyle \frac{{y}^{2}}{4}}.$ 
This manipulation is called completing the square [1] in ${x}^{2}+xy$, or completing the square ${x}^{2}$.
Replacing $y$ by $y$, we also have
$${x}^{2}xy={\left(x\frac{y}{2}\right)}^{2}\frac{{y}^{2}}{4}.$$ 
Here are some applications of this method:

•
http://planetmath.org/DerivationOfQuadraticFormulaDerivation of the solution formula to the quadratic equation.

•
Putting the general equation of a circle, ellipse, or hyperbola into standard form, e.g. the circle
${x}^{2}+{y}^{2}+2x+4y=5\Rightarrow {(x+1)}^{2}+{(y+2)}^{2}=10,$ from which it is frequently easier to read off important information (the center, radius, etc.)

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Completing the square can also be used to find the extremal value of a quadratic polynomial [2] without calculus. Let us illustrate this for the polynomial^{} $p(x)=4{x}^{2}+8x+9$. Completing the square yields
$p(x)$ $=$ ${(2x+2)}^{2}4+9$ $=$ ${(2x+2)}^{2}+5$ $\ge $ $5,$ since ${(2x+2)}^{2}\ge 0$. Here, equality holds if and only if $x=1$. Thus $p(x)\ge 5$ for all $x\in \mathbb{R}$, and $p(x)=5$ if and only if $x=1$. It follows that $p(x)$ has a global minimum^{} at $x=1$, where $p(1)=5$.

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Completing the square can also be used as an integration technique to integrate, for example the function $\frac{1}{4{x}^{2}+8x+9}$ [1].
References
 1 R. Adams, Calculus, a complete^{} course, AddisonWesley Publishers Ltd, 3rd ed.
 2 Matematiklexikon (in Swedish), J. Thompson, T. Martinsson, Wahlström & Widstrand, 1991.
(Anyone has an English reference?)
Title  completing the square 

Canonical name  CompletingTheSquare 
Date of creation  20130322 13:36:27 
Last modified on  20130322 13:36:27 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  14 
Author  mathcam (2727) 
Entry type  Algorithm 
Classification  msc 00A20 
Related topic  SquareOfSum 