Table of Contents

About

In mathematics,

  • a quadratic function,
  • a quadratic polynomial,
  • a polynomial of degree 2,
  • or simply a quadratic,

is a polynomial function in one or more variables in which the highest-degree term is of the second degree.

Quadratus is the Latin word for square.

Type

Univariate

Single variable

<math>f(x)=ax^2+bx+c,\quad a \ne 0</math>

In elementary algebra, such polynomials often arise in the form of a quadratic equation <math>ax^2 + bx + c = 0</math> . The solutions to this equation are called the roots of the quadratic polynomial, and may be found through:

  • factorization,
  • completing the square,
  • graphing,
  • Newton's method,
  • or through the use of the quadratic formula.

Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.

Bivariate

The bivariate case in terms of variables x and y has the form

<math> f(x,y) = a x^2 + by^2 + cx y+ d x+ ey + f \,\!</math>

Formula / Solution

The quadratic formula <MATH> x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\ \ </MATH> expresses the solution of the quadratic equation <MATH>ax^2 + bx + c = 0</MATH>

The symbol <math>\pm</math> expresses the fact that there is two soluctions (ie two root).

Plot

Example: Plot of <math>0.5 (x-1) (x-4)</math> A quadratic function with roots x = 1 and x = 4.

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Documentation / Reference