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mcastillo356

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- I've been searching in the net; I understand the whole quote I will write down; but the concept, the words "complete the square" keep me wondering if they really relate to a geometric and well known shape, or it is just an algebraic term.

Hi, PF

"

There is a well-known formula for finding the roots of a quadratic polynomial.

The two solutions of the quadratic equation

##Ax^2+Bx+C=0##,

where ##A##, ##B##, and ##C## are constants and ##A\neq{0}##, are given by

$$x=\displaystyle\frac{-B\pm\sqrt{B^2-4AC}}{2A}$$

To see this, just divide the equation by ##A## and complete the square for the terms in ##x##

##x^2+\displaystyle\frac{B}{A}x+\displaystyle\frac{C}{A}=0##

##x^2+\displaystyle\frac{2B}{2A}x+\displaystyle\frac{B^2}{4A^2}=\displaystyle\frac{B^2}{4A^2}-\displaystyle\frac{C}{A}##

##\Bigg(x+\displaystyle\frac{B}{2A}\Bigg)^2=\displaystyle\frac{B^2-4AC}{4A^2}##

##x+\displaystyle\frac{B}{2A}=\pm\displaystyle\frac{\sqrt{B^2-4AC}}{2A}##."

(i)- There is an algebraic set out of a square when stating ##\Bigg(x+\displaystyle\frac{B}{2A}\Bigg)^2##: this is an squared expression.

(ii)- The above expression, from a geometric point of view, is actually an square (height per equal width).

Am I right?

Greetings!

"

**Roots and Factors of Quadratic Polynomials**There is a well-known formula for finding the roots of a quadratic polynomial.

**The Quadratic Formula**The two solutions of the quadratic equation

##Ax^2+Bx+C=0##,

where ##A##, ##B##, and ##C## are constants and ##A\neq{0}##, are given by

$$x=\displaystyle\frac{-B\pm\sqrt{B^2-4AC}}{2A}$$

To see this, just divide the equation by ##A## and complete the square for the terms in ##x##

##x^2+\displaystyle\frac{B}{A}x+\displaystyle\frac{C}{A}=0##

##x^2+\displaystyle\frac{2B}{2A}x+\displaystyle\frac{B^2}{4A^2}=\displaystyle\frac{B^2}{4A^2}-\displaystyle\frac{C}{A}##

##\Bigg(x+\displaystyle\frac{B}{2A}\Bigg)^2=\displaystyle\frac{B^2-4AC}{4A^2}##

##x+\displaystyle\frac{B}{2A}=\pm\displaystyle\frac{\sqrt{B^2-4AC}}{2A}##."

*Attempt*: The words "complete the square" for the terms in ##x## are both an algebraic and a geometric ideas in the proof of the two solutions for the quadratic equation:(i)- There is an algebraic set out of a square when stating ##\Bigg(x+\displaystyle\frac{B}{2A}\Bigg)^2##: this is an squared expression.

(ii)- The above expression, from a geometric point of view, is actually an square (height per equal width).

Am I right?

Greetings!