Definition:Quadratic Equation

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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.


Canonical Form

The canonical form of the quadratic equation is:

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

where $a$, $b$ and $c$ are constants.


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: $x^2 + 1 = 0$

The quadratic equation:

$x^2 + 1 = 0$

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

$x = \pm i$

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


Example: $x^2 + 4 = 0$

The quadratic equation:

$x^2 + 4 = 0$

has the wholly imaginary roots:

$x = \pm 2 i$

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

  • Results about quadratic equations can be found here.


Historical Note

The ancient Babylonians knew the technique of solving quadratic equations as long ago as $1600$ BCE.

The ancient Greeks, a thousand years or so later, solved quadratics by geometric constructions.

The general algebraic formulation of its solution did not appear until at least $100$ CE.


Sources

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