Completing the square

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

x 2 + b x

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

Now, consider the problem of finding this antiderivative:

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

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