Search for a tool
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.

Results

Completing the Square -

Tag(s) : Symbolic Computation

Share
Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!


Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


Feedback and suggestions are welcome so that dCode offers the best 'Completing the Square' tool for free! Thank you!

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 equivalent 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 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 $.

Why using the completion of a square?

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.

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, breaker, 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!
Reminder : dCode is free to use.

Cite dCode

The copy-paste of the page "Completing the Square" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Completing the Square on dCode.fr [online website], retrieved on 2024-12-03, https://www.dcode.fr/square-completion

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Completing the Square' tool for free! Thank you!


https://www.dcode.fr/square-completion
© 2024 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.
 
Feedback