Completing the square
|
|
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.