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 ?
To his credit, Khayyam also postulated that a cubic equation could have more than one solution, demonstrating that equations could have more than one solution, though he did not advance the notion that a cubic could have three solutions.
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2 the exact solution of the cubic equation (3) is contrasted with the approximations (5) allowing for different powers of [epsilon].
Feroiu, "Calculation of Joule-Thomson Inversion Curves from a General Cubic Equation of State," Fluid Phase Equilibria.
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Nevertheless, Figure 1A shows a curve derived by fitting a cubic equation.
This is obvious by a glance to the plot, but could be rigorously proven upon transporting the origin to the inflection point and then noticing that the new cubic equation does not change upon replacing the positive values by the negative values along both the horizontal and vertical axis.
The Treatise on Algebra describes Khayyam's solution to the cubic equation, for which he employed several ingenious curve constructions.
The amount of rotation information is then used by the vi to generate a velocity profile according to the cubic equation stated above.
From Elphick's work, Swartzendruber and Barber (1965) developed mathematically a cubic equation and tested this hypothesis.
The following cubic equation is suggested for the calculation of the riser diameter for steel castings: