# Cubic Equation

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## cubic equation

[′kyü·bik i′kwā·zhən]## Cubic Equation

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

*ax ^{3} + bx^{2} + cx + d* = 0

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

*y*^{3} + *py* + *q* = 0

where

*p* = *b*^{2}/3*a*^{2} = *c*/*a*

*q* = *2b*^{2}/27*a*^{3} − *bc*/*3a*^{2} + *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 *q ^{2}/4 + p/27* in Cardan’s formula. If

*q*+

^{2}/4*p*> 0, then the cubic equation has three different roots, one real and two complex conjugates. If

^{3}/4 + p^{3}/27*q*+

^{2}/4*p*= 0, then all three roots are real, two of them being equal. If

^{3}27*q*+

^{2}/4*p*

^{3}/27 > 0, then the three roots are real and different. The expression

*q*+

^{2}/4*p*differs by a constant factor from the discriminant of a cubic equation

^{3}/27*D = −4p*.

^{3}− 27q^{2}### 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.