Cubic Equation (redirected from Cardano formulae)

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 = 2b²/27a³ − bc/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.