PlanetMath: completing the square

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completing the square

(Algorithm)

Let us consider the expression , where and are real (or complex) numbers. Using the formula

$\displaystyle (x+y)^2 = x^2+2xy +y^2$

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

, or completing the square

Replacing by , we also have

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

Here are some applications of this method:

Derivation 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

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

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 . Completing the square yields

$\displaystyle p(x)$ $\displaystyle =$ $\displaystyle (2x+2)^2-4 +9$

$\displaystyle =$ $\displaystyle (2x+2)^2+5$

$\displaystyle \ge$ $\displaystyle 5,$

since $(2x+2)^2\ge 0$ . Here, equality holds if and only if . Thus $p(x)\ge 5$ for all $x\in \mathbb{R}$ , and if and only if . It follows that has a global minimum at , where .
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].

Bibliography

1: R. Adams, Calculus, a complete course, Addison-Wesley Publishers Ltd, 3rd ed.
2: Matematik Lexikon (in Swedish), J. Thompson, T. Martinsson, Wahlström & Widstrand, 1991.

(Anyone has an English reference?)

"completing the square" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Cross-references: function, integration technique, global minimum, equality, Calculus, polynomial, radius, center, information, hyperbola, ellipse, circle, equation, complex, real, expression
There are 6 references to this entry.

This is version 10 of completing the square, born on 2003-05-02, modified 2007-05-09.
Object id is 4237, canonical name is CompletingTheSquare.
Accessed 18443 times total.

Classification:

00A20 (General :: General and miscellaneous specific topics :: Dictionaries and other general reference works)

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