Quadratic function

A quadratic function of one variable has the form

a x^{2} + b x + c

with

a \neq = 0

.

Table of contents
- Quadratic equation Quadratic function
  - Quadratic formula Quadratic equation
  - Quadratic inequality Quadratic equation

Quadratic equation

The roots of

a x^{2} + b x + c = 0

, where

a \neq = 0

, are

x = \frac{- b \pm b ^{2} - 4 a c}{2 a} .

Quadratic formula ( $x = (- b \pm b^{2} - 4 a c) / (2 a)$ )

For a quadratic equation

a x^{2} + b x + c = 0

with

a \neq = 0

, completing the square gives

x = \frac{- b \pm b ^{2} - 4 a c}{2 a} .

Over the complex numbers both roots exist, with a repeated root of a polynomial when the discriminant

b^{2} - 4 a c

is zero. Over the real numbers the discriminant determines whether there are two distinct roots, one repeated root, or no real root.

Quadratic inequality

For a real quadratic

a s^{2} + b s + c

with

a > 0

, strict negativity occurs between its two distinct real roots, if they exist. Completing the square gives discriminant

b^{2} - 4 a c > 0

as the existence condition. A physical domain restriction must be intersected with that interval.

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A quadratic function is a type of polynomial function of the form: \[ f(x) = ax^2 + bx + c \] where: - \( a \), \( b \), and \( c \) are constants (with \( a \neq 0 \)), - \( x \) is the variable, - \( a \) determines the direction of the parabola (if \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it

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