HP50g simplifing a root
10-06-2020, 12:06 PM (This post was last modified: 10-10-2020 11:42 AM by Albert Chan.)
Post: #16
 Albert Chan Senior Member Posts: 2,682 Joined: Jul 2018
RE: HP50g simplifing a root
I discovered a necessary condition to simplify cubic: ³√(A ± √R) = a ± √r
If we found a, we can get r = (A/a-a*a)/3

Simplify equations from find_cbrt(): 4a³ - A = (3a) ³√(A²-R)

This implied ³√(A²-R) must be integer !

Example, ³√(9416 - 4256√5)

A² - R = 9416² - 4256² * 5 = -1906624
c = ³√(A² - R) = -124 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ // cube root may be simplified

4a³ = A + 3*c*a
a = ³√(A/4 + (3c/4)*a)

With my Casio FX 115MS:

5 =
³√(2354 - 93 Ans
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 12.36167494
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 10.63945257
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 11.09160665
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 10.97648027 // locking to 11, try it
11 = ↑
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 11 ﻿ ﻿ ﻿ ﻿ ﻿ // = a
9416 / Ans - Ans Ans
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 735 ﻿ ﻿ ﻿ // = 3r
√(Ans / 3 / 5
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 7 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ // = b

Add the sign, ³√(9416 - 4256√5) = 11 - 7√5
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 Messages In This Thread HP50g simplifing a root - peacecalc - 09-29-2020, 09:22 PM RE: HP50g simplifing a root - Albert Chan - 09-29-2020, 11:47 PM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 02:22 AM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 10:50 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 07:31 AM RE: HP50g simplifing a root - peacecalc - 09-30-2020, 05:33 AM RE: HP50g simplifing a root - peacecalc - 10-01-2020, 02:20 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 05:22 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 06:05 PM RE: HP50g simplifing a root - Albert Chan - 10-04-2020, 11:48 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 07:36 PM RE: HP50g simplifing a root - peacecalc - 10-05-2020, 11:36 AM RE: HP50g simplifing a root - Albert Chan - 10-05-2020, 05:01 PM RE: HP50g simplifing a root - peacecalc - 10-06-2020, 05:25 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 09:40 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020 12:06 PM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 04:13 PM RE: HP50g simplifing a root - Albert Chan - 10-07-2020, 06:12 PM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 12:20 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 02:31 PM RE: HP50g simplifing a root - Albert Chan - 10-11-2020, 06:28 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 03:17 AM RE: HP50g simplifing a root - Albert Chan - 10-24-2020, 02:19 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 10:54 PM RE: HP50g simplifing a root - CMarangon - 10-12-2020, 11:45 PM RE: HP50g simplifing a root - grsbanks - 10-13-2020, 06:46 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 05:21 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 03:58 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 04:49 PM RE: HP50g simplifing a root - peacecalc - 10-12-2020, 08:49 PM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:30 AM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:36 AM

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