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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

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

### 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 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 $$n-1$$.

## Source code

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