Completing the Square

Question 1

What is a square completion? (Definition)

Answer

Completion of the square is the name given to a method of factorization of the polynomials due to this degree. Factoring takes its name from the fact that the equivalent factored form obtained has the variable in a squared expression.

Question 2

How to complete the square?

Answer

dCode can complete the square and find factors by depressing a polynomial expression

A quadratic polynomial $ x^2 +bx + c = 0 $ can be modified by adding $ (b/2)^2 - c - (b/2)^2 + c (= 0) $ that allows factorizing in $$ (x +(b/2))^2 - (b/2)^2 + c $$

Example: $ p(x)=2x^2+12x+14 $, in order to complete the square hand, factorize by the coefficient of $ x^2 $ (here $ 2 $): $ p(x)=2(x^2+6x+7) $ and continue with polynomial $ q(x) = x^2+6x+7 $

Example: Identify the coefficient of $ x $, here $ 6 $ and divide it by $ 2 $ to get $ β=6/2=3 $ and use $ β $ to write $$ q(x) = x^2 + 6x + 7 = (x+β)^2 − β^2 + 7 = (x+3)^2 − 2 $$

Example: Back to $ p(x) = 2q(x) $ to get the completed square: $$ p(x) = 2x^2 + 12x + 14 = 2 ( (x+3)^2 − 2 ) = 2 (x+3)^2 − 4 $$

With the factorized form, it becomes simple to find the roots.

$$ p(x) = 0 \iff 2(x+3)^2−6 = 0 \iff (x+3)^2 = 3 \\ \iff x+3 = \pm \sqrt{3} \iff x = \pm \sqrt{3} - 3 $$

dCode can generalize this approach to other polynomials of order $ n > 2 $ by removing the term of degree $ n-1 $.

Question 3

Why using the completion of a square?

Answer

Square completion is used to simplify quadratic polynomial expressions by factorization. This factorization makes it possible to find the roots of the polynomial and therefore to solve equations more easily.

Completing the Square