Mathematics - Quadratic (function|polynomial of degree 2)

In mathematics,

• a polynomial of degree 2,

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

$f(x)=ax^2+bx+c,\quad a \ne 0$

In elementary algebra, such polynomials often arise in the form of a quadratic equation $ax^2 + bx + c = 0$ . 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

$f(x,y) = a x^2 + by^2 + cx y+ d x+ ey + f \,\!$

Formula / Solution

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

The symbol $\pm$ expresses the fact that there is two soluctions (ie two root).

Plot

Example: Plot of $0.5 (x-1) (x-4)$ A quadratic function with roots x = 1 and x = 4.