I The solution to a cubic equation

Charles Link

Homework Helper
Insights Author
Gold Member
2018 Award
4,345
1,833
Summary
The solution of ## x^3-x+.1=0 ## is of interest. A quick approximation to the 3 real roots is found. This is compared with the general solution that is found to the equation in the form ## x^3+px=q ##, using the substitution ## x=w-\frac{p}{3w} ##, (the standard way of solving this type of equation), and obtaining a quadratic in ## w^3 ##.
The equation ## x^3-x+.1=0 ##, has 3 real roots that can be quickly approximated as follows: Writing the equation ## x=.1+x^3 ##, iterative methods quickly indicate that there is a root ## r_1 ## near ## x=+.1 ##, and more accurately ## r_1 \approx + .101 ##. ## \\## Doing an approximate calculation of factoring out ## x-.1 ## from ## x^3-x+.1 ## yields ## (x-.1)(x^2+.1x-1 )=0 ## (approximately). ## \\ ## The quadratic formula can be quickly applied to the quadratic factor to give ## r_2 \approx +.95 ## and ## r_3 \approx -1.05 ##. ## \\ ## These results were found to be in agreement with the standard solution to this cubic equation using Vieta's substitution ## x=w-\frac{p}{3w} ##. ## \\ ## Here ## p=-1 ##, and ## q=-.1 ##, for the equation in the form ## x^3+px=q ##. ## \\ ## The substitution gives a quadratic in ## w^3 ## that is ## w^6-qw^3-\frac{p^3}{27}=0 ##. ## \\ ## The number ## w^3 \approx .192 e^{i (\theta +n 360^{\circ}) } ## where ## \theta \approx 105.1^{\circ} ## and ## \theta \approx 254.9^{\circ} ##, and ## n=0,1,2 ##. ## \\ ## The complex number ## w ## is then found as ## w= (.192)^{1/3} e^{i (\theta/3+n 120^{\circ})}=.577 e^{i (\theta/3+n 120^{\circ})} ##. ## \\ ## There are only 3 distinct ## w's ##. ## \\ ## When the computation is then made for ## x ## from ## w ##, the result is ## x=w+\frac{1}{3w}\approx 2(.577) \cos{\phi} ## , and ## x ## is real as it must be. The three roots ## x \approx .101, .95, -1.05 ## are then obtained as above. ## \\ ## Note: I could have kept more decimals in these calculations, but I did them mostly by hand, using Taylor series expansions, etc. to get them reasonably precise. The goal here was simply to show agreement with the very quick solution above.
 
Last edited:

Charles Link

Homework Helper
Insights Author
Gold Member
2018 Award
4,345
1,833
One comment on the above: I did this problem mostly for my own interest. I wanted to come up with a cubic equation where iterative methods would supply the first root. The other two roots can then be quickly found (approximately) by factoring out ## x-r_1 ##, and solving the quadratic expression. ## \\ ## I had to google the method for the general solution to the cubic equation. I had seen it in high school, (at least one method of solution), but that was about 45 years ago for me. ## \\ ## One additional comment I have that perhaps someone can give some feedback=I made the statement that there are only 3 distinct ## w's ##: I think there might be 6 distinct ##w's ##. The pair of ##w's ## with ## \phi=##35 degrees ## =105/3 ## and ## 325 ## degrees ##=255/3+240 ## give the same root ## x=+.95 ##, but they are distinct. The quadratic in ## w ^3 ## is a 6th power equation, so it comes as no surprise that there could be 6 different ## w's ##. ## \\ ## Likewise, ## \phi= 105/3+120=155^{\circ} ## and ## 255/3+120=205^{\circ} ## both form the x=-1.05 root, while 105/3+240=275 and 255/3=85 give the x=+.101 root. ## \\ ## I think we can conclude there are 6 distinct ## w's ##.## \\## [Edit: The quadratic equation for ## w^3 ## gets two separate solutions for ## w^3=re^{i \theta} ##, (two different ## \theta's ##), and in subsequently solving for ## w ##, the integer ## n ## above gives 3 distinct values for each ## \theta ##, for ## n=0, 1, 2 ##. (Of course ## n ## can be any integer, but only ## n=0,1,2 ## give distinct values for ## w ##.)]
 
Last edited:

Want to reply to this thread?

"The solution to a cubic equation" You must log in or register to reply here.

Related Threads for: The solution to a cubic equation

Replies
5
Views
651
  • Posted
Replies
3
Views
1K
Replies
11
Views
3K
Replies
2
Views
2K
Replies
2
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top