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# Quadratic polynomials

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

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

## Roots of a quadratic equation and the quadratic formula

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

${\displaystyle {1}}$

with ${\displaystyle {1}}$ and ${\displaystyle {1}}$.

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

${\displaystyle {1}}$

or, letting ${\displaystyle {1}}$,

${\displaystyle {1}}$

hence

${\displaystyle {1}}$

yielding the quadratic formula

${\displaystyle {1}}$

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

• 0 (in which case ${\displaystyle {1}}$ 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).

## Vieta's formulas for the quadratic

Vieta's formulas for the quadratic

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

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

${\displaystyle {1}}$