Completing the Square

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 factored form obtained has the variable in a squared expression.

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

A quadratic polynomial $ x^2 +bx + c = 0 $ can be modified in $ (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 the coefficient of $ x^2 $ : $ p(x)=2(x^2+6x+7) $ and continue with $ 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 $$ q(x) = x^2 + 6x + 7 = (x+3)^2 − β^2 + 7 = (x+3)^2 − 2 $$

Example: Back to $ p(x) = 2q(x) $ to get the completed square: $$ p(x)=2x^2+12x+14=2((x+3)^2−2)=2(x+3)^2−6 $$

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 $.