root, in mathematics, a solution to an equation, usually expressed as a number or an algebraic formula.

In the 9th century, Arab writers usually called one of the equal factors of a number jadhr (“root”), and their medieval European translators used the Latin word radix (from which derives the adjective radical). If a is a positive real number and n a positive integer, there exists a unique positive real number x such that xn = a. This number—the (principal) nth root of a—is written nSquare root of a or a1/n. The integer n is called the index of the root. For n = 2, the root is called the square root and is written Square root of a . The root 3Square root of a is called the cube root of a. If a is negative and n is odd, the unique negative nth root of a is termed principal. For example, the principal cube root of –27 is –3.

If a whole number (positive integer) has a rational nth root—i.e., one that can be written as a common fraction—then this root must be an integer. Thus, 5 has no rational square root because 22 is less than 5 and 32 is greater than 5. Exactly n complex numbers satisfy the equation xn = 1, and they are called the complex nth roots of unity. If a regular polygon of n sides is inscribed in a unit circle centred at the origin so that one vertex lies on the positive half of the x-axis, the radii to the vertices are the vectors representing the n complex nth roots of unity. If the root whose vector makes the smallest positive angle with the positive direction of the x-axis is denoted by the Greek letter omega, ω, then ω, ω2, ω3, …, ωn = 1 constitute all the nth roots of unity. For example, ω = −1/2 + Square root of −3 /2, ω2 = −1/2 − Square root of −3 /2, and ω3 = 1 are all the cube roots of unity. Any root, symbolized by the Greek letter epsilon, ε, that has the property that ε, ε2, …, εn = 1 give all the nth roots of unity is called primitive. Evidently the problem of finding the nth roots of unity is equivalent to the problem of inscribing a regular polygon of n sides in a circle. For every integer n, the nth roots of unity can be determined in terms of the rational numbers by means of rational operations and radicals; but they can be constructed by ruler and compasses (i.e., determined in terms of the ordinary operations of arithmetic and square roots) only if n is a product of distinct prime numbers of the form 2h + 1, or 2k times such a product, or is of the form 2k. If a is a complex number not 0, the equation xn = a has exactly n roots, and all the nth roots of a are the products of any one of these roots by the nth roots of unity.

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The term root has been carried over from the equation xn = a to all polynomial equations. Thus, a solution of the equation f(x) = a0xn + a1xn − 1 + … + an − 1x + an = 0, with a0 ≠ 0, is called a root of the equation. If the coefficients lie in the complex field, an equation of the nth degree has exactly n (not necessarily distinct) complex roots. If the coefficients are real and n is odd, there is a real root. But an equation does not always have a root in its coefficient field. Thus, x2 − 5 = 0 has no rational root, although its coefficients (1 and –5) are rational numbers.

More generally, the term root may be applied to any number that satisfies any given equation, whether a polynomial equation or not. Thus π is a root of the equation x sin (x) = 0.

This article was most recently revised and updated by William L. Hosch.
Key People:
Bhāskara II
Herbert Westren Turnbull
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quadratic equation, in mathematics, an algebraic equation of the second degree (having one or more variables raised to the second power). Old Babylonian cuneiform texts, dating from the time of Hammurabi, show a knowledge of how to solve quadratic equations, but it appears that ancient Egyptian mathematicians did not know how to solve them. Since the time of Galileo, they have been important in the physics of accelerated motion, such as free fall in a vacuum. The general quadratic equation in one variable is ax2 + bx + c = 0, in which a, b, and c are arbitrary constants (or parameters) and a is not equal to 0. Such an equation has two roots (not necessarily distinct), as given by the quadratic formula x = b ± b 2 4 a c 2 a

The discriminant b2 − 4ac gives information concerning the nature of the roots (see discriminant). If, instead of equating the above to zero, the curve ax2 + bx + c = y is plotted, it is seen that the real roots are the x coordinates of the points at which the curve crosses the x-axis. The shape of this curve in Euclidean two-dimensional space is a parabola; in Euclidean three-dimensional space it is a parabolic cylindrical surface, or paraboloid.

In two variables, the general quadratic equation is ax2 + bxy + cy2 + dx + ey + f = 0, in which a, b, c, d, e, and f are arbitrary constants and a, c ≠ 0. The discriminant (symbolized by the Greek letter delta, Δ) and the invariant (b2 − 4ac) together provide information as to the shape of the curve. The locus in Euclidean two-dimensional space of every general quadratic in two variables is a conic section or its degenerate.

More general quadratic equations, in the variables x, y, and z, lead to generation (in Euclidean three-dimensional space) of surfaces known as the quadrics, or quadric surfaces.

The Editors of Encyclopaedia Britannica This article was most recently revised and updated by William L. Hosch.