Expert: Paul Klarreich - 11/26/2009

Question
QUESTION: Hello!

My problem is the following equation:

8x^3-6x+1=0

Coud you help me to solve it?

Thank you

ANSWER: Questioner: grull
Country: Hungary
Category: Advanced Math
Subject: third degrees equation
Question: Hello!

My problem is the following equation:

8x^3-6x+1=0

Coud you help me to solve it?

Thank you
.........................................
Sorry, but there are no rational solutions and the formula for finding irrational ones is too complex.

All you can do is approximate the solutions numerically.


---------- FOLLOW-UP ----------

QUESTION: Hi!

I know about Cardano's formula, but my solution goes always wrong (i don't know exactly, why). But i've already found the "best" root: x=sin (pi/18) (if we use sin 3x=4sinx-3sin^3 x then it's done)

God bless you!

Answer
Questioner: grull
Country: Hungary
Category: Advanced Math
Private: Yes
Subject: Third degree equation
Question: QUESTION: Hello!

My problem is the following equation:

8x^3-6x+1=0

Coud you help me to solve it?

Thank you

ANSWER: Questioner: grull
Country: Hungary
Category: Advanced Math
Subject: third degrees equation
Question: Hello!

My problem is the following equation:

8x^3-6x+1=0

Coud you help me to solve it?

Thank you
.........................................
Sorry, but there are no rational solutions and the formula for finding irrational ones is too complex.

All you can do is approximate the solutions numerically.

---------- FOLLOW-UP ----------

QUESTION: Hi!

I know about Cardano's formula, but my solution goes always wrong (i don't know exactly, why). But i've already found the "best" root: x=sin (pi/18) (if we use sin 3x=4sinx-3sin^3 x then it's done)

God bless you!

Yes, that does work out.  Your identity for sin(3x) is correct, and

4 sin (pi/18) - 3 sin^3(pi/18) = sin(pi/6)

4 sin (pi/18) - 3 sin^3(pi/18) = 1/2, so

8 sin (pi/18) - 6 sin^3(pi/18) - 1 = 0

I knew someone who claimed to know Cardano's formula (I just know it exists).  Alas, he committed suicide at a young age and I never got to check out his solutions.

BE WARNED!