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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
$\displaystyle x^{2}+xy$  $\displaystyle=$  $\displaystyle x^{2}+xy+0$  
$\displaystyle=$  $\displaystyle x^{2}+xy+\frac{y^{2}}{4}\frac{y^{2}}{4}$  
$\displaystyle=$  $\displaystyle\left(x+\frac{y}{2}\right)^{2}\frac{y^{2}}{4}.$ 
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:

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 $p(x)=4x^{2}+8x+9$. Completing the square yields
$\displaystyle p(x)$ $\displaystyle=$ $\displaystyle(2x+2)^{2}4+9$ $\displaystyle=$ $\displaystyle(2x+2)^{2}+5$ $\displaystyle\geq$ $\displaystyle 5,$ since $(2x+2)^{2}\geq 0$. Here, equality holds if and only if $x=1$. Thus $p(x)\geq 5$ 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 $p(1)=5$.

Completing the square can also be used as an integration technique to integrate, for example the function $\displaystyle\frac{1}{4x^{2}+8x+9}$ [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?)
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