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#1 2006-01-03 15:32:08

God
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Registered: 2005-08-25
Posts: 59

A cubic...

Can the cubic 0 = x^3 - x - 1 be solved by means of simple algebra (without the cubic formula)?

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#2 2006-01-03 16:36:40

ganesh
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Registered: 2005-06-28
Posts: 13,235

Re: A cubic...

x³ - x - 1 = 0
x³ - x = 1
x(x²-1)=1
x = 1/(x²-1)
Does this cubic equation have a real solution? sad


Character is who you are when no one is looking.

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#3 2006-01-03 17:14:59

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: A cubic...

No, it can't be factored.  The only real factor is a real number x~1.32.  So any attempts to factor it would be futile.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2006-01-03 23:03:44

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

It has one real root, but I used the cubic formula:
x≈1.32471795724474602596090885448...


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#5 2006-01-04 02:50:21

Ricky
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Registered: 2005-12-04
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Re: A cubic...

So what is the exact root of it, krassi?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2006-01-04 03:03:31

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

No. Do you REALLY want it?


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#7 2006-01-04 10:33:23

Ricky
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Registered: 2005-12-04
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Re: A cubic...

If you use the cubic formula, shouldn't you come up with the exact root?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#8 2006-01-04 16:26:50

ganesh
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Registered: 2005-06-28
Posts: 13,235

Re: A cubic...

I did not know the roots of a cubic equation and looked for it on the net. I got to this pdf. Ricky, you can find the root of the incomplete cubic equation using the formulae.


Character is who you are when no one is looking.

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#9 2006-01-04 22:07:58

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

The roots of quadratic equation ax^2+bx+c:

View Image: ^2.GIF

Last edited by krassi_holmz (2006-01-04 22:08:31)


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#10 2006-01-04 22:09:13

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

Cubic equation:

View Image: ^3.GIF

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#11 2006-01-04 22:10:17

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

And quadric equation (useless):

View Image: ^4r1.GIF View Image: ^4r2.GIF View Image: ^4r3.GIF View Image: ^4r4.GIF

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#12 2006-01-04 22:12:05

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

For you question we get:

View Image: ueq.GIF

Last edited by krassi_holmz (2006-01-04 22:12:44)


IPBLE:  Increasing Performance By Lowering Expectations.

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#13 2006-01-04 22:13:30

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

Equations >4 are unsolvable exactin radicals. (Galoa)


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#14 2006-01-04 22:16:31

krisper
Member
Registered: 2005-12-20
Posts: 19

Re: A cubic...

I am sure you could find this without using the cubic formula.
x^3 - x - 1 = 0
x(x^2-1) = 1
if x = 0, the equasion has no meaning, so x must differ from 0 (x<>0); Now we can devide by x.
x^2 - 1 = 1/x.
We draw the graphics of these two expressions - x^2 - 1 and 1/x. After that we check where they cross eachother. This happens only in I quadrant which means that x^3 - x - 1 = 0 has only one real root. Afterwards doing some calculas we find that the root is somewhere between 1 and √2. Now we have to use the tangents and make some calculations and I am sure we will get the exact value of this root.

Last edited by krisper (2006-01-04 22:16:42)


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#15 2006-01-04 22:25:50

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

this is the exact root!

krassi_holmz wrote:

For you question we get:


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#16 2006-01-05 16:34:50

ganesh
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Registered: 2005-06-28
Posts: 13,235

Re: A cubic...

Very good posts and images uploaded, krassi_holmz. Those would be very useful. BTW, did you know the insolubility of the quintic equation was shown first by Neils Henrik Abel, the Norwegain Mathematician?


Character is who you are when no one is looking.

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#17 2006-01-05 22:21:40

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

Yes, I've read very much about Abel.

And the Galous theory simplifies the result.
The solubility of an equation depends of the structude if its Galous group.


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#18 2006-01-05 22:23:49

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

The general quintic can be solved in terms of Jacobi theta functions, as was first done by Hermite  in 1858. Kronecker  subsequently obtained the same solution more simply, and Brioschi also derived the equation.


IPBLE:  Increasing Performance By Lowering Expectations.

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#19 2006-01-08 05:21:32

God
Member
Registered: 2005-08-25
Posts: 59

Re: A cubic...

Lol.thanks... wow... that looks so complicated lol

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#20 2006-01-08 05:30:07

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: A cubic...

That's why I prefer the approximated result.


IPBLE:  Increasing Performance By Lowering Expectations.

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#21 2006-01-08 09:41:47

God
Member
Registered: 2005-08-25
Posts: 59

Re: A cubic...

I guess you might as well just use Newton's recursion for these things...

Last edited by God (2006-01-08 09:41:58)

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