How can I effectively use my mathematical knowledge to solve complex equations?

  • Context: Undergrad 
  • Thread starter Thread starter minusequals
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around solving the intersection of a cubic function and a circle, specifically the equations y=x^3 and x^2+y^2=9. Participants explore various mathematical tools and approaches, including analytical and numerical methods, to find solutions to the equations presented.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant inquires about the intersection of a cubic and a circle, mentioning Descartes' rule of signs and the rational zero theorem as initial tools.
  • Another participant notes that there is no general analytic solution for sextic equations and suggests transforming the sextic equation into a cubic by letting u = x^2.
  • A different participant confirms the transformation to a cubic and suggests numerical solutions or using software like Maple for explicit formulas, providing a complex expression as a potential solution.
  • One participant claims to have calculated a numerical solution to the cubic, arriving at an approximate value of 1.3855 for the intersection point.
  • There is a light-hearted exchange regarding the time available for solving the problem, with one participant expressing a desire to solve it by hand.
  • Another participant humorously comments on a quote from Spiderman, leading to a brief off-topic discussion about comic knowledge.

Areas of Agreement / Disagreement

Participants express differing preferences for solving the equations, with some favoring analytical methods while others suggest numerical approaches. The discussion does not reach a consensus on the best method to use.

Contextual Notes

The discussion includes various mathematical transformations and methods, but it does not resolve the complexities involved in solving the equations or the efficacy of the proposed approaches.

Who May Find This Useful

Individuals interested in mathematical problem-solving, particularly those dealing with intersections of functions and numerical methods, may find this discussion relevant.

minusequals
Messages
1
Reaction score
0
where do a cubic and a circle intersect? y=x^3, x^2+y^2=9? Descartes' rule of signs and rational zero theorem at least tell me we have a symmetical +- irrational root; what tools do I use next? is sin theta = cos^3 theta going anywhere? x^2 + x^6 - 9 = 0. I have seen a graphical solution, but have no idea which analytical tools to use.
 
Mathematics news on Phys.org
There is, in general, no analytic solution for sextic equations (this has been proven). For this one in particular, we have... x^{6} + x^{2} - 9 = 0 Let u = x^{2}, giving us...u^{3} + u - 9 Now approach it as a cubic.
 
By letting u = x2 you get a cubic in u. While it is possible to solve cubics such as this by hand, you likely have better things to do. You could solve it numerically or use a program such as Maple to write out an explicit formula. Here's the exact answer given by Maple for the positive value of x where they intersect (hard to read and parse):

(1/6)*sqrt(6)*sqrt((972+12*sqrt(6573))^(1/3)*((972+12*sqrt(6573))^(2/3)-12))/(972+12*sqrt(6573))^(1/3)

or approximately 1.385703836.
 
Last edited by a moderator:
LCKurtz said:
By letting u = x2 you get a cubic in u. While it is possible to solve cubics such as this by hand, you likely have better things to do.
Well, I don't! Using the cubic formula (and, I will admit, a calculator) I got 1.9195 as a solution to the cubic, to four decimal places. The square root of that is 1.3855, basically your answer. Using that you can reduce the problem to a quadratic.

You could solve it numerically or use a program such as Maple to write out an explicit formula. Here's the exact answer given by Maple for the positive value of x where they intersect (hard to read and parse):

(1/6)*sqrt(6)*sqrt((972+12*sqrt(6573))^(1/3)*((972+12*sqrt(6573))^(2/3)-12))/(972+12*sqrt(6573))^(1/3)

or approximately 1.385703836.
LCKurtz, I accidently clicked on "edit" when I intended "quote". Sorry about that. I think I put your post back together correctly.
 
LCKurtz said:
While it is possible to solve cubics such as this by hand, you likely have better things to do...

HallsofIvy said:
Well, I don't!

Heh. Nothing else to do eh? :cool:
LCKurtz, I accidently clicked on "edit" when I intended "quote". Sorry about that. I think I put your post back together correctly.

"With great power comes great responsibility." -- Spiderman
 
LCKurtz said:
Heh. Nothing else to do eh? :cool:


"With great power comes great responsibility." -- Spiderman

Pfft...he was only quoting his Aunt. Your knowledge of Marvel comics is insufficient for you to be giving advice on this forum.
 

Similar threads

Undergrad The solution to a cubic equation
  • · Replies 2 ·
Replies
2
Views
2K
High School Are Inverse Problems More Complex Than Direct Problems in Mathematics?
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad Can New Theorems Predict Incomplete 5x5 Magic Square Solutions?
  • · Replies 68 ·
3
Replies
68
Views
12K
Undergrad Quadratic, cubic, quartic, quintic equations
  • · Replies 16 ·
Replies
16
Views
5K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
Undergrad Solving simultaneous vector equations
  • · Replies 11 ·
Replies
11
Views
3K
MHB Using Trigonometry for solving a Cubic equation
  • · Replies 5 ·
Replies
5
Views
4K
How should I use the averaging approximation to find this?
  • · Replies 3 ·
Replies
3
Views
2K
MHB How can I factor a polynomial without using the rational root theorem?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Solve for x⁴+16x-12=0 using effective methods
  • · Replies 8 ·
Replies
8
Views
3K