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 ight)-(b/2)^2 = (x+(b/2))^2-b^2/4.

See quadratic equation.

= 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:

:intfrac{dx}{9x^2-90x+241}.

The denominator is

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

Adding (10/2)2 = 25 to x 2 - 10 x gives a perfect square x 2 - 10 x + 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

:intfrac{dx}{9x^2-90x+241}=frac{1}{9}intfrac{dx}{(x-5)^2+(4/3)^2}=frac{1}{9}cdotfrac{3}{4}arctanfrac{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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