# 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:

• Putting the general equation of a circle, ellipse, or hyperbola into standard form, e.g. the circle

 $\displaystyle x^{2}+y^{2}+2x+4y=5\Rightarrow(x+1)^{2}+(y+2)^{2}=10,$

from which it is frequently easier to read off important information (the center, radius, etc.)

• 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, Addison-Wesley Publishers Ltd, 3rd ed.
• 2 Matematiklexikon (in Swedish), J. Thompson, T. Martinsson, Wahlström & Widstrand, 1991.

(Anyone has an English reference?)

Title completing the square CompletingTheSquare 2013-03-22 13:36:27 2013-03-22 13:36:27 mathcam (2727) mathcam (2727) 14 mathcam (2727) Algorithm msc 00A20 SquareOfSum