Definition

A quadratic equation is a polynomial equation of the form:

$a x^2 + b x + c = 0$

such that $a \ne 0$.

From Solution to Quadratic Equation, the solutions are:

$x = \dfrac {-b \pm \sqrt {b^2 - 4 a c} } {2 a}$

Discriminant

The expression $b^2 - 4 a c$ is called the discriminant of the equation.

Also defined as

Some older treatments of this subject present this as:

An algebraic equation of the form $a x^2 + 2 b x + c = 0$ is called a quadratic equation.
It has solutions:
$x = \dfrac {-b \pm \sqrt {b^2 - a c} } a$

but this approach has fallen out of fashion.

Also known as

A quadratic equation is also known as:

• an equation of the second degree
• a polynomial of degree $2$

and so on.

Examples

Example: $x^2 + 1 = 0$

$x^2 + 1 = 0$

has no root in the set of real numbers $\R$:

$x = \pm i$

where $i = \sqrt {-1}$ is the imaginary unit.

Example: $z^2 - \paren {3 + i} z + 4 + 3 i = 0$

The quadratic equation in $\C$:

$z^2 - \paren {3 + i} z + 4 + 3 i = 0$

has the solutions:

$z = \begin{cases} 1 + 2 i \\ 2 - i \end{cases}$

Example: $z^2 + \paren {2 i - 3} z + 5 - i = 0$

The quadratic equation in $\C$:

$z^2 + \paren {2 i - 3} z + 5 - i = 0$

has the solutions:

$z = \begin{cases} 2 - 3 i \\ 1 + i \end{cases}$

Also see

The ancient Babylonians knew the technique of solving quadratic equations as long ago as $1600$ BCE.
The general algebraic formulation of its solution did not appear until at least $100$ CE.