A while ago I've posted a program that uses *ACOSH* to solve a quadratic equation:

Short Quadratic Solver (HP-42S) Message #29

Recently I stumbled across a way to solve a cubic equation using trigonometry. Some of you might find that interesting as well.

Cheers

Thomas

## Equations

A general cubic equation can be transformed to the following form using a substitution:

The trick is to use the following identity:

Here we set:

With the following substitution we can get a closed-form solution:

A similar formula can be found using the following identity:

## Programs

00 { 31-Byte Prgm } 00 { 34-Byte Prgm }

01>LBL "CuEq" 01>LBL "CuEqH"

02 2 02 2

03 / 03 /

04 X<>Y 04 X<>Y

05 -3 05 3

06 / 06 /

07 / 07 /

08 LASTX 08 LASTX

09 SQRT 09 SQRT

10 / 10 /

11 LASTX 11 LASTX

12 X<>Y 12 X<>Y

13 ASIN 13 ASINH

14 3 14 3

15 / 15 /

16 SIN 16 SINH

17 * 17 *

18 2 18 -2

19 * 19 *

20 END 20 END

## Examples

x^{3} + 6x - 2 = 0

CuEq: 3.27480002074E-1 i0

CuEqH: 3.27480002074E-1

x^{3} + 3x - 4 = 0

CuEq: 1 i0

CuEqH: 1

## References

*Edited: 22 Mar 2011, 12:51 p.m. after one or more responses were posted*