Completing the square - Definition and Overview

Completing the square is a technique of elementary algebra wherein an expression

<math>x^2+bx<math>

is replaced by one of the form

<math>(x+c)^2+d.<math>

Specifically, we have

<math>\left(x^2+bx+(b/2)^2\right)-(b/2)^2 = (x+(b/2))^2-b^2/4.<math>

See quadratic equation.

Example

A simple example is this.

<math>x^2+4x = (x+2)^2-c = (x^2+4x+4)-4<math>

Now, consider the problem of finding this antiderivative:

<math>\int\frac{dx}{9x^2-90x+241}.<math>

The denominator is

<math>9x^2-90x+241=9(x^2-10x)+241.<math>

Adding (10/2)² = 25 to x² - 10x gives a perfect square x² - 10x + 25 = (x - 5)². So we get

<math>9(x^2-10x)+241=9(x^2-10x+25)+241-9(25)=9(x-5)^2+16.<math>

Our integral becomes

<math>\int\frac{dx}{9x^2-90x+241}=\frac{1}{9}\int\frac{dx}{(x-5)^2+(4/3)^2}=\frac{1}{9}\cdot\frac{3}{4}\arctan\frac{3(x-5)}{4}+C.<math>

Completing the square reduces any problem involving a quadratic polynomial to one involving a square quadratic polynomial and a constant.

External link

• Completing the square (http://planetmath.org/?op=getobj&from=objects&id=4237) on PlanetMath.