Quadratic polynomial

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Quadratic polynomial function of two independent variables x and y.

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2. For example, x2 − 4x + 7 is a quadratic polynomial, while x3 − 4x + 7 is not.


The coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring.


When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "degenerate case". Usually the context will establish which of the two is meant.

Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial.


A quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z.

The one-variable case

Any single-variable quadratic polynomial may be written as

where x is the variable, and a, b, and c represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation . 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.

If the polynomial is a polynomial in one variable, it determines a quadratic function in one variable. An example is given by f(x) = x2 + x − 2;. The graph of such a function is a parabola (in degenerate cases a line), and its zeroes can be found by solving the quadratic equation f(x) = 0.

There are three main forms :

Two variables case

Any quadratic polynomial with two variables may be written as

where x and y are the variables and a, b, c, d, e, and f are the coefficients. Such polynomials are fundamental to the study of conic sections. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces. In linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space.

N variables case

In the general case, a quadratic polynomial in n variables x1, ..., xn can be written in the form

where Q is a symmetric n-dimensional matrix, P is an n-dimensional vector, and R a constant.

See also


  • Algebra 1, Glencoe, ISBN 0-07-825083-8
  • Algebra 2, Saxon, ISBN 0-939798-62-X


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