Univariate Second-Order Polynomial 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]\displaystyle{ ax^2+bx+c=0 }[/math] with [math]\displaystyle{ a≠0 }[/math] where a, b and c are constant coefficients and [math]\displaystyle{ 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]\displaystyle{ ax^2+bx+c=0 }[/math] are the points at which the graph of [math]\displaystyle{ 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]\displaystyle{ a \gt 0 }[/math], the parabola opens upwards; when [math]\displaystyle{ a \lt 0 }[/math], the parabola opens downwards the axis of symmetry is the line [math]\displaystyle{ x= -b/2a }[/math].
  • In context of numerical analysis, the quadratic equation can be expressed as the roots of the polynomial [math]\displaystyle{ P(x) }[/math] [math]\displaystyle{ 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]\displaystyle{ n=2 }[/math], this is simply [math]\displaystyle{ P(x)=a_2 x^2 + a_{1}x+ a_0 = 0 }[/math] Thus, comparing to the general quadratic equation given above [math]\displaystyle{ a_2=a }[/math], [math]\displaystyle{ a_1=b }[/math] and [math]\displaystyle{ a_0=c }[/math]
• Example(s):
  • [math]\displaystyle{ 1x^2 + 0x + -1 = 0 }[/math], or [math]\displaystyle{ x^2 - 1 = 0 }[/math].
  • [math]\displaystyle{ 4x^2 + -8x + 3 = 0 }[/math].
  • [math]\displaystyle{ 3.4x^2 + 2.1x + 9.6 = 0 }[/math].
  • …
• Counter-Example(s):
  • a Cubic Equation.
  • an Exponential Equation.
  • a Linear Equation.
• See: Quadratic Function, Quadratic Scaling, Graphical Solutions of a Quadratic Equation.

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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]\displaystyle{ 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.

1999

• http://mathworld.wolfram.com/QuadraticEquation.html :
  • A quadratic equation is a second-order polynomial equation in a single variable x: [math]\displaystyle{ ax^2+bx+c=0,\, }[/math] with a [math]\displaystyle{ ≠ 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.

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