Cubic Equation


Also found in: Dictionary, Wikipedia.
Related to Cubic Equation: quadratic equation

cubic equation

[′kyü·bik i′kwā·zhən]
(mathematics)
A polynomial equation with no exponent larger than 3.

Cubic Equation

 

an algebraic equation of the third degree. The general form of a cubic equation is

ax3 + bx2 + cx + d = 0

where a ≠ 0. By replacing x in this equation by a new unknown y related to x by x = y − b/3a, a cubic equation can be reduced to the simpler (canonical) form

y3 + py + q = 0

where

p = b2/3a2 = c/a

q = 2b2/27a3bc/3a2 + da

The solution of this equation can be found using Cardan’s formula

If the coefficients of a cubic equation are real, then the nature of its roots depends on the sign of the radicand q2/4 + p/27 in Cardan’s formula. If q2/4 + p3/4 + p3/27 > 0, then the cubic equation has three different roots, one real and two complex conjugates. If q2/4 + p327 = 0, then all three roots are real, two of them being equal. If q2/4 + p3/27 > 0, then the three roots are real and different. The expression q2/4 + p3/27 differs by a constant factor from the discriminant of a cubic equation D = −4p3 − 27q2.

REFERENCES

Kurosh, A. G. Kurs vysshei algebry, 9th ed. Moscow, 1968.
Entsiklopediia elementarnoi matematiki, book 2. [Edited by P. S. Aleksandrov (et al.).] Moscow-Leningrad, 1951.
References in periodicals archive ?
Our comparison also showed that the median PSA for Puerto Rican men with proven PC followed a cubic equation in time (Fig.
As a corollary of Theorem 3, we obtain the following stability result of cubic equation (6) associated with cubic Lie derivation on the Banach algebra [[chi].sub.[rho]], which may be considered as endowed with modular [rho] = [parallel] x [parallel].
Here [bar.x] = [[bar.x].sub.1]/[[bar.x].sub.2] is a real positive root of the cubic equation
There may be more than one solutions to the one-element cubic equation and the roots obtained by the Newton-Simpson iterative method are correlated to the initial values.
[29] Aavatsmark I, Kometa BK, Gasda SE, Sandve TH, Nilsen HM (2015) A Generalized Cubic Equation of State with Application to Pure CO2 Injection in Aquifers.
However, it may be useful to pose the question, "Is there a mathematical limit to how much a system should be oversized to reduce power used at a given design airflow?" The power used by an oversized unit in Equation 21 was the product of the part-load power fraction (a cubic equation) that decreased with increased oversizing and a second term on the right-hand side of Equation 21, which was a linear term that increased with increased oversizing.
Beginning with the Renaissance solution of the cubic equation, he traces developments to Galois' description of his ideas, encountering all the important concepts, each in a well-motivated manner.
3) Cubic interpolation algorithm, where is computed the interpolated values in point X_interp, Y_interp; between neighboring points given by cubic equation;
Specifically we showed that the roots of a simple cubic equation accurately yield the eigenfrequencies of a 2D PC constructed with polar crystals.
Moreover, any solution z of the original cubic equation is given by the Cardano Formula [alpha](p, q) - 3p/ [alpha](p, q) (see [2]).
Taking a closer look to these Figures it is concluded that unlike cubic equation of state that cannot predict the critical region of VLE data, the ANN has well results in this region.