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In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is The general form of a quartic equation is Graph of a polynomial function of degree 4, with its 4 roots and 3 critical points .
Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus.It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics.
Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.
The values of trigonometric functions of angles related to / satisfy cubic equations. Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic. The solution of the general quartic equation relies on the solution of its resolvent cubic.
In some cases, the concept of resolvent cubic is defined only when P(x) is a quartic in depressed form—that is, when a 3 = 0. Note that the fourth and fifth definitions below also make sense and that the relationship between these resolvent cubics and P ( x ) are still valid if the characteristic of k is equal to 2 .
The cruciform curve, or cross curve is a quartic plane curve given by the equation = where a and b are two parameters determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation, x ↦ 1/x, y ↦ 1/y to the ellipse a 2 x 2 + b 2 y 2 = 1, and is therefore a rational plane algebraic curve of genus zero.
In mathematics, the term quartic describes something that pertains to the "fourth order", such as the function . It may refer to one of the following: Quartic function, a polynomial function of degree 4; Quartic equation, a polynomial equation of degree 4; Quartic curve, an algebraic curve of degree 4
This table is complete for odd primes less than 50. To check whether a number m is a quadratic residue mod one of these primes p, find a ≡ m (mod p) and 0 ≤ a < p. If a is in row p, then m is a residue (mod p); if a is not in row p of the table, then m is a nonresidue (mod p).