# 1911 Encyclopædia Britannica/Equation/Cubic Equations

*Cubic Equations*.

1. Cubic equations, like all equations above the first degree, are
divided into two classes: they are said to be *pure* when they contain
only one power of the unknown quantity; and *adfected* when they
contain two or more powers of that quantity.

Pure cubic equations are therefore of the form *x*^{3} = *r*; and hence
it appears that a value of the simple power of the unknown quantity
may always be found without difficulty, by extracting the cube root
of each side of the equation. Let us consider the equation *x*^{3} − *c*^{3} = 0
more fully. This is decomposable into the factors *x* − *c* = 0 and
*x*^{2} + *cx* + *c*^{2} = 0. The roots of this quadratic equation are 12 (−1 ± √−3) c,
and we see that the equation *x*^{3} = *c*^{3} has three roots, namely, one real
root c, and two imaginary roots 12 (−1 ± √−3) c. By making *c* equal
to unity, we observe that 12 (−1 ± √−3) are the imaginary cube roots
of unity, which are generally denoted by ω and ω^{2}, for it is easy to
show that (12 (−1 − √−3))^{2} = 12 (−1 + √−3).

2. Let us now consider such cubic equations as have all their terms, and which are therefore of this form,

*x*

^{3}+ A

*x*

^{2}+ B

*x*+ C = 0,

where A, B and C denote known quantities, either positive or negative.

This equation may be transformed into another in which the second
term is wanting by the substitution *x* = *y* − A/3. This transformation is
a particular case of a general theorem. Let *x*^{n} + A*x*^{n−1} + B*x*^{n−2} ... = 0.
Substitute *x* = *y* + h; then (*y* + *h*)^{n} + A (*y* + *h*)^{n−1} ... = 0. Expand each
term by the binomial theorem, and let us fix our attention on the
coefficient of *y*^{n−1}. By this process we obtain 0 = *y*^{n} + *y*^{n−1}(A + *nh*) +
terms involving lower powers of *y*.

Now *h* can have any value, and if we choose it so that A + *nh* = 0,
then the second term of our derived equation vanishes.

Resuming, therefore, the equation *y*^{3} + *qy* + *r* = 0, let us suppose
*y* = *v* + *z*; we then have *y*^{3} = *v*^{3} + *z*^{3} + 3*vz* (*v* + *z*) = *v*^{3} + *z*^{3} + 3*vzy*, and the
original equation becomes *v*^{3} + *z*^{3} + (3*vz* + *q*) *y* + *r* = 0. Now *v* and *z*
are any two quantities subject to the relation *y* = *v* + *z*, and if we
suppose 3*vz* + *q* = 0, they are completely determined. This leads to
*v*^{3} + *z*^{3} + *r* = 0 and 3*vz* + *q* = 0. Therefore *v*^{3} and *z*^{3} are the roots of the
quadratic *t*^{2} + *rt* − *q*^{2}/27 = 0. Therefore

v^{3} = | −12 r + √(127 q^{3} + 14 r^{2}); z^{3} = −12 r − √(127 q^{3} + 14r^{2}); |

v = | ∛{−12 r + √(127 q^{3} + 14 r^{2}) }; z = ∛{ (−12 r − √(127 q^{3} + 14 r^{2}) }; |

and y = | v + z = ∛{−12 r + √(127q^{3} + 14 r^{2}) } + ∛{−12 r − √(127 q^{3} + 14 r^{2}) }. |

Thus we have obtained a value of the unknown quantity *y*, in terms
of the known quantities *q* and *r*; therefore the equation is resolved.

3. But this is only one of three values which *y* may have. Let us,
for the sake of brevity, put

*r*+ √(127

*q*

^{3}+ 14

*r*

^{2}), B = −12

*r*− √(127

*q*

^{3}+ 14

*r*

^{2}),

and put | α = 12 (−1 + √−3), |

β = 12 (−1 − √−3). |

Then, from what has been shown (§ 1), it is evident that *v* and *z* have
each these three values,

*v*= ∛A,

*v*= α∛A,

*v*= β∛A;

*z*= ∛B,

*z*= α∛B,

*z*= β∛B.

To determine the corresponding values of *v* and *z*, we must consider
that *vz* = −13 *q* = ∛(AB). Now if we observe that αβ = 1, it will
immediately appear that *v* + *z* has these three values,

*v*+

*z*= ∛A + ∛B,

*v* + *z* = α∛A + β∛B,

*v*+

*z*= β∛A + α∛B,

which are therefore the three values of *y*.

The first of these formulae is commonly known by the name of
Cardan’s rule (see Algebra: *History*).

The formulae given above for the roots of a cubic equation may
be put under a different form, better adapted to the purposes of
arithmetical calculation, as follows:—Because *vz* = −13 *q*, therefore
*z* = −13*q* × 1/*v* = −13 *q* / ∛A; hence *v* + *z* = ∛A − 13 *q* / ∛A: thus it appears
that the three values of *y* may also be expressed thus:

*y*= ∛A − 13

*q*/ ∛A

*y* = α∛A − 13 *q*β / ∛A

*y*= β∛A − 13

*q*α / ∛A.

See below, *Theory of Equations*, §§ 16 et seq.