Quadratic polynomials

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A [univariate] quadratic polynomial is a [univariate] polynomial of degree 2, i.e. of the form

ax^{2}+bx+c,\quad a\neq 0.

Roots of a quadratic equation and the quadratic formula

The two zeros of the quadratic polynomial ${1}$ are the two roots of the quadratic equation

{1}

with ${1}$ and ${1}$ .

The two roots are obtained by completing the square, i.e.

{1}

or, letting ${1}$ ,

{1}

hence

{1}

{1}

where ${1}$ , the discriminant of the quadratic equation, is either:

0 (in which case ${1}$ is the rational double root of the quadratic equation);
positive and a perfect square (the quadratic equation has two distinct rational roots);
positive and not a perfect square (the quadratic equation has two distinct real conjugate quadratic roots);
negative (the quadratic equation has two distinct complex conjugate quadratic roots).

(x-x_{1})(x-x_{2})=0,\,

gives a system of two equations in two variables (which are the two roots)

{1}