Plane trigonometry, by S.L. Loney.

SOLUTION OF A CUBIC EQUATION. 469
Now (2) and (3) are the same equation if
3 1
z= cos 0, 3pn =, and - cos 30 = qn3.
Hence n= (4p7
and therefore cos 30 =-4q(p.................. (4).
The equation (4) can always be solved (by means of
the tables if necessary) if
p be positive, and 4q ( < 1,
i.e. if q2 < 4p3.
[The student who is acquainted with the Theory of Equations will
notice that is the case which cannot be solved by Cardan's Method. It
is the case when the roots of the original cubic are all real. ]
If 0 be the smallest angle satisfying equation (4), then
2w 47r
the values 0+ -and 0 +
3 3
also satisfy it, so that the roots of the equation
3 - 3px + q =0
1 1 /27\
are - cos 0, cos 0 + -, and cos 0 +
n n 3 n 33/
i. e. 2 /p cos 0, 2/p cos (0+ 27), and 2 ^p cos ( +4).
398. Ex. Solve the equation
x3+6x2+9x+3=0.
Put x=y- 2, and the equation becomes
y - 3y + 1=0.