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In mathematics, a quadratic function of a single variable is a function of the form [1] = + +,,where is its variable, and , , and are coefficients.The expression + + , especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two.
Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
Functions of the form = have at most one -intercept, but may contain multiple -intercepts. The x {\displaystyle x} -intercepts of functions, if any exist, are often more difficult to locate than the y {\displaystyle y} -intercept, as finding the y {\displaystyle y} -intercept involves simply evaluating the function at x = 0 {\displaystyle x=0} .
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is + + =, where a ≠ 0. The quadratic equation on a number can be solved using the well-known quadratic formula, which can be derived by completing the square.
That is, h is the x-coordinate of the axis of symmetry (i.e. the axis of symmetry has equation x = h), and k is the minimum value (or maximum value, if a < 0) of the quadratic function. One way to see this is to note that the graph of the function f ( x ) = x 2 is a parabola whose vertex is at the origin (0, 0).
For example, a degree two polynomial in two variables, such as + +, is called a "binary quadratic": binary due to two variables, quadratic due to degree two. [ a ] There are also names for the number of terms, which are also based on Latin distributive numbers, ending in -nomial ; the common ones are monomial , binomial , and (less commonly ...
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).