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

Completing the Square - dCode

Tag(s) : Symbolic Computation

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*!

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

**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 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 the approach to other polynomials of order \( n \) superior to 2 by removing the term of degree \( n-1 \).

dCode retains ownership of the source code of the script Completing the Square online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Completing the Square script for offline use on PC, iPhone or Android, ask for price quote on contact page !

completion,complete,square,factorization,factor,factorize,polynomial,quadratic

Source : https://www.dcode.fr/square-completion

© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode

Feedback

▲