# Cubic Equation

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## 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 ?
For this assessment, the UN structure for the G matrix and the quadratic model with intercept, the cubic model without intercept, and the linear model with intercept were used (Table 2).
The square (quadratic) model in state a and the special cubic model in state ab have been yielded for the root width b_kor, Table 2.
Tables 5-6 show the analysis of variance of the rheological parameters [[tau].sub.o], K, and n at 20 and 60[degrees]C, fitted according to the quadratic or cubic models. Analysis of variance (P-test) showed that the second-order (quadratic and cubic) models fitted well to the experimental data and the only exception was the flow behaviour index at 60[degrees]C where the p value was not significant at 5% of probability.
The cubic model, they explain, was "among the two best fitting
Collectively, the comparative results for the linear, quadratic, and cubic models for the full data set demonstrate that the cubic model provides the best description of the relationship between the Human Freedoms Index and the Corruption Perceptions Index.
The estimated coefficients for the cubic model with industry effects included are found in column 3 of Table 1.
We observe a 2% improvement of quadratic and cubic model over the linear model.
Table 5 shows that the adjusted [R.sup.2] increases from the linear to the quadratic model and from the quadratic to the cubic model. The Ramsey RESET test was performed to check if the increase in adjusted [R.sup.2] is sufficient to reject the linear and quadratic models in favor of the cubic.
Another good fit of our model occurs at the bottom of the Dark Ages, predicted by our cubic model to occur around the year 932 A.
These models are the Overlap Model (Blumenthal & Kupfer, 1986), the Three Element Model (Jacobs, Brewer, & Klein-Benham, 1999), the Suicide Trajectory Model (Stillion, McDowell, & May, 1989), and the Cubic Model (Shneidman, 1987).
The [R.sup.2]s were: .550 for the quadratic model, .566 for the log model, and .594 for the cubic model. All three [R.sup.2]s were higher than the .390 of the linear model.

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