HP50g simplifing a root

[Albert Chan]

(10-09-2020 02:31 PM)Albert Chan Wrote:  Lets rearrange the cubic to match form x³ + 3px - 2q = 0

c³ = A² - R
4*a³ = 3*c*a + A
a³ + 3*(-c/4)*a - 2*(A/8) = 0

→ Cubic discriminant = p³ + q² = (-c/4)³ + (A/8)² = (-c³ + A²) / 64 = R/64
→ If R < 0, we got 3 real roots.

Instead of cubic discriminant, we can show this with more familiar quadratic discriminant.

Let f(x) = 4x³ - 3*c*x - A, so that f(a) = 0

f(x) = 4 * (x-a) * (x² + a*x + (a² - 3*c/4))

Quadratic discriminant = a² - 4*(a² - 3*c/4) = -3*(a² - c) = -3*r
→ if r < 0 (due to R < 0), f got 3 real roots.

We can solve the quadratics, but unnecessary.
If we have 1 solution, where (a ± √r)³ = A ± √R, we can get the other 2.

(a ± √r)³ = ((a ± √r) × w)³ = ((a ± √r) / w)³,     where w = (-1)^(2/3)

Example, simplify ³√(36 + 20i√7)

>>> from mpmath import *
>>> A, B, k = 36, 20, -7
>>> z = A + B*sqrt(k)
>>> q = cbrt(z); print q
(3.79128784747792 + 1.27520055582102j)

>>> c = cbrt(A*A-B*B*k); print c
16.0

q looks hopeless to simplify, but c is integer, thus possible.
Perhaps the "nice" root is not the principle root ?

>>> w = exp(2j/3*pi)
>>> print q*w, q/w
(-3.0 + 2.64575131106459j)      (-0.791287847477919 - 3.92095186688561j)

>>> a = -3
>>> f = lambda x: 4*x**3 - 3*c*x - A
>>> print f(a)
0.0
>>> print a, (A/a - a*a)/3
-3       -7.0

We have ³√(36 + 20i√7) = (-3 + i√7) / (-1)^(2/3)

However, converted to form a ± √r, expression look messier than original ³√(A + √R)