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

  • Thread starter minusequals
  • Start date
In summary, the conversation discusses the intersection of a cubic and a circle, the use of Descartes' rule of signs and rational zero theorem, and the solution for a sextic equation using analytical tools. The participants also mention the use of numerical or programmatic methods for solving the equation. They also have a playful exchange about the responsibilities that come with knowledge and the importance of accuracy in quoting sources.
  • #1
minusequals
1
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
  • #2
There is, in general, no analytic solution for sextic equations (this has been proven). For this one in particular, we have... [tex]x^{6} + x^{2} - 9 = 0[/tex] Let [itex]u = x^{2}[/itex], giving us...[tex]u^{3} + u - 9[/tex] Now approach it as a cubic.
 
  • #3
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:
  • #4
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.
 
  • #5
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
 
  • #6
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.
 

1. What is a sextic equation?

A sextic equation is a polynomial equation of degree six, meaning it contains a term with a variable raised to the sixth power.

2. What is the general form of a sextic equation?

The general form of a sextic equation is ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g = 0, where a, b, c, d, e, f, and g are constants and x is the variable.

3. How do I solve a sextic equation using algebraic methods?

Solving a sextic equation using algebraic methods can be a lengthy and complex process. It typically involves using techniques such as factoring, substitution, and the quadratic formula. However, it is not always possible to find exact solutions for a sextic equation using algebraic methods.

4. Are there other methods for solving sextic equations?

Yes, there are other methods for solving sextic equations, such as using numerical methods like Newton's method or the bisection method. These methods involve approximating the solutions of the sextic equation rather than finding exact solutions.

5. Are there any real-world applications for sextic equations?

Yes, sextic equations have various applications in fields such as physics, engineering, and economics. For example, they can be used to model the motion of objects under the influence of gravity or to find the optimal solution for a production cost function.

Similar threads

I Geometry and algebraic equations
    • General Math
Replies
4
Views
1K
B Calculating the perfect tennis shot using vectors
    • General Math
Replies
2
Views
1K
I The solution to a cubic equation
    • General Math
Replies
2
Views
1K
MHB Time-Discrete Odometry: Solving Complicated Equations By Hand
    • General Math
Replies
2
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
226
B Invariant under rotation: Banal, obvious, or noteworthy?
    • General Math
Replies
2
Views
1K
Find v_1 for projectile using start position, final position and angle
    • Introductory Physics Homework Help
Replies
10
Views
892
MHB How can I factor a polynomial without using the rational root theorem?
    • General Math
Replies
3
Views
1K
Can B-splines cause oscillations in second derivatives?
    • General Math
Replies
2
Views
1K
MHB Using Trigonometry for solving a Cubic equation
    • General Math
Replies
5
Views
4K

Hot Threads

Recent Insights

Back
Top