completing the square

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

x^2+bx

is replaced by one of the form

(x+c)^2+d.

Specifically, we have

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

Example

A simple example is this.

x^2+4x = (x+2)^2-c = (x^2+4x+4)-4

Now, consider the problem of finding this antiderivative:

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

The denominator is

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

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

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

Our integral becomes

\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.

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

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