Completing the square: Meaning (information, definition, explanation, facts)

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

x² + bx

is replaced by one of the form

(x + c)² + d.

Specifically, we have

See quadratic equation.

Example

A simple example is this.

x² + 4x = (x + 2)² - c = (x² + 4x + 4) - 4

Now, consider the problem of finding this antiderivative:

The denominator is

9x² - 90x + 241 = 9(x² - 10x) + 241.

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

9(x² - 10x) + 241 = 9(x² - 10x + 25) + 241 - 9(25) = 9(x - 5)² + 16.

Our integral becomes

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