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So, if the three non-monic coefficients of the depressed quartic equation, + + + =, in terms of the five coefficients of the general quartic equation are given as follows: =, = + and = +, then the criteria to identify a priori each case of quartic equations with multiple roots and their respective solutions are exposed below.
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.
A quartic equation has four solutions, and only one solution for this equation matches the problem as presented. Another solution is for a case where one ladder (and wall) is below ground level and the other above ground level. In this case the ladders do not actually cross, but their extensions do so at the specified height.
An algebraic solution to the problem was finally found first in 1965 by Jack M. Elkin (an actuary), by means of a quartic polynomial. [8] Other solutions were rediscovered later: in 1989, by Harald Riede; [9] in 1990 (submitted in 1988), by Miller and Vegh; [10] and in 1992, by John D. Smith [3] and also by Jörg Waldvogel. [11]
Quartic reciprocity, a theorem from number theory; Quartic surface, a surface defined by an equation of degree 4; See also. All pages with titles beginning with Quartic ; All pages with titles containing Quartic; Quart (disambiguation) Quintic, relating to degree 5, as next higher above quartic; Cubic (disambiguation), relating to degree 3 or a ...
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An affine quartic surface is the solution set of an equation of the form (,,) =
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]
The solution of the general quartic equation relies on the solution of its resolvent cubic. The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix.