# Definition:Quadratic Equation

## 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 9$: Solutions of Algebraic Equations: $9.1$: Quadratic Equation - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.10$: Quadratic equations - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Polynomial Equations: $31$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**quadratic (quadric)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**quadratic** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $4$: Lure of the Unknown: Algebra - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**quadratic equation**