# Univariate Second-Order Polynomial Equation

(Redirected from quadratic equation)

A Univariate Second-Order Polynomial Equation is a univariate equation that is a second-order polynomial equation.

**AKA:**Quadratic Equation.**Context:**- It can be expressed as [math]ax^2+bx+c=0[/math] with [math]a≠0[/math] where a, b and c are constant coefficients and [math]x[/math] is a single numeric variable.
- It can be an input to a Quadratic Equation Solving Task (that produces quadratic equation solutions).
- It can have two real solutions ... [math]ax^2+bx+c=0[/math] are the points at which the graph of [math]y=ax^2+bx+c[/math] intersects the x-axis.
- It can, in the context of algebraic geometry, correspond to the standard form of a parabola equation in Cartesian coordinates with following characteristics: when [math] a \gt 0 [/math], the parabola opens upwards; when [math] a \lt 0 [/math], the parabola opens downwards the axis of symmetry is the line [math] x= -b/2a [/math].
**In context of numerical analysis**, the quadratic equation can be expressed as the roots of the polynomial [math]P(x)[/math] [math]P(x)=a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0 = 0[/math] for [math]n=2[/math], this is simply [math]P(x)=a_2 x^2 + a_{1}x+ a_0 = 0[/math] Thus, comparing to the general quadratic equation given above [math]a_2=a[/math], [math]a_1=b[/math] and [math]a_0=c[/math]

**Example(s):**- [math]1x^2 + 0x + -1 = 0[/math], or [math]x^2 - 1 = 0[/math].
- [math]4x^2 + -8x + 3 = 0[/math].
- [math]3.4x^2 + 2.1x + 9.6 = 0[/math].

**Counter-Example(s):****See:**Quadratic Function, Quadratic Scaling, Graphical Solutions of a Quadratic Equation.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/quadratic_equation Retrieved:2015-11-2.
- In elementary algebra, a
**quadratic equation**(from the Latin*quadratus*for “square") is any equation having the form : [math] ax^2+bx+c=0 [/math] where*x*represents an unknown, and*a*,*b*, and*c*represent known numbers such that*a*is not equal to 0. If*a*0, then the equation is linear, not quadratic. The numbers*a*,*b*, and*c*are the*coefficients*of the equation, and may be distinguished by calling them, respectively, the*quadratic coefficient*, the*linear coefficient*and the*constant*or*free term*.Because the quadratic equation involves only one unknown, it is called “univariate”. The quadratic equation only contains powers of

*x*that are non-negative integers, and therefore it is a polynomial equation, and in particular it is a second degree polynomial equation since the greatest power is two.Quadratic equations can be solved by a process known in American English as factoring and in other varieties of English as

*factorising*, by completing the square, by using the quadratic formula, or by graphing. Solutions to problems equivalent to the quadratic equation were known as early as 2000 BC.

- In elementary algebra, a

### 1999

- http://mathworld.wolfram.com/QuadraticEquation.html :
- A quadratic equation is a second-order polynomial equation in a single variable x: [math] ax^2+bx+c=0,\, [/math] with a [math]≠ 0.[/math]. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex.