HP50g simplifing a root

[Albert Chan]

(10-10-2020 03:58 PM)Albert Chan Wrote:  Substitute back in, many terms get cancelled, a = A / (c²/d - c + d)

If ³√(A ± √R) can be simplified, c turns to integer (we had shown this earlier).
In fact, the whole denominator turns to integer = a² + 3r = 4a² - 3c

Proof is trivial:

if ³√(A ± √R) = a ± √r, then

c = ³√(A² - R) = a² - r
d = ³√(|A|+√R)² = (|a|+√r)²

c²/d + d - c = (|a|-√r)² + (|a|+√r)² - c = (2a² + 2r) - (a² - r) = a² + 3r