Polynomial Equation

Polynomial Equation has general form

${\displaystyle A_{n}x^{n}+A_{n-1}x^{n-1}+A_{1}x^{1}+A_{o}x^{0}=0}$

1st ordered polynomial equation

1st ordered polynomial equation general form

${\displaystyle Ax+B=0}$

Divide the quadratic equation by a, which is allowed because a is non-zero:

${\displaystyle x+{\frac {B}{A}}=0}$

${\displaystyle x=-{\frac {B}{A}}}$

In summary,

1st ordered polynomial equation ${\displaystyle Ax+B=0}$ has root ${\displaystyle x=-{\frac {B}{A}}}$

2nd ordered polynomial equation

2nd ordered polynomial equation has general form

${\displaystyle Ax^{2}+Bx+C=0}$

The quadratic formula can be derived with a simple application of technique of completing the square.Divide the quadratic equation by a, which is allowed because a is non-zero:

${\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0.}$

Subtract c/a from both sides of the equation, yielding:

${\displaystyle x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}.}$

The quadratic equation is now in a form to which the method of completing the square can be applied. Thus, add a constant to both sides of the equation such that the left hand side becomes a complete square:

${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2},}$

which produces:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}.}$

Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.}$

The square has thus been completed. Taking the square root of both sides yields the following equation:

${\displaystyle x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}.}$

Isolating x gives the quadratic formula:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.}$

Summary

Polynomial Equation	Equation	Root
1st ordered equation	${\displaystyle ax+b=0}$	${\displaystyle x=-{\frac {b}{a}}}$
2nd ordered equation	${\displaystyle ax^{2}+bx+c=0}$	${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.}$