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Completing the Square

Tool to make automatic square completion. Square completing is a calculation method allowing to factor a quadratic polynomial expression using the polynomial depression method.

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Completing the Square -

Tag(s) : Symbolic Computation

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Completing the Square

Completing the square solver


Answers to Questions (FAQ)

What is a square completion? (Definition)

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.

How to complete the square?

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 $.

Source code

dCode retains ownership of the "Completing the Square" source code. Except explicit open source licence (indicated Creative Commons / free), the "Completing the Square" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "Completing the Square" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Completing the Square" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
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Cite as source (bibliography):
Completing the Square on dCode.fr [online website], retrieved on 2022-12-09, https://www.dcode.fr/square-completion

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