# Root (mathematics)

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In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that

f(x) = 0.

For an important special case see zero (complex analysis).

Consider the function f defined by the following formula:

${\displaystyle f(x)=x^{2}-6x+9\,}$

Now 3 is called a root of f, because f(3) = 32 - (6 x 3) + 9 = 0.

If the function is mapping from real numbers to real numbers, its zeros are essentially where its graph hits the x-axis. In this situation, the root can be called a x-intercept. Although, not all graphs cross the x-axis and in these cases the root is a complex number, where it is a multiple of the root of negative one[ -1]. Complex roots may only occur in pairs and indeed linear graphs never have complex roots.

The word root can also refer to a number in the form x1/a, such as the square root or other roots.

A substantial amount of mathematics was developed in order to find roots of various functions, especially polynomials. One wide-ranging concept, complex numbers, was developed to handle the roots of quadratic equations with negative discriminant (that is, those leading to expressions involving the square root of negative numbers).

All real polynomials of odd degree have a real number as a root. Many real polynomials of even degree do not have a real root, but the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of real polynomials come in conjugate pairs.

One of the most important unsolved problems in mathematics concerns the location of the roots of the Riemann zeta function.