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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

### 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

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Completing the Square on dCode.fr [online website], retrieved on 2024-09-13, https://www.dcode.fr/square-completion

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