xtremee Posted November 3, 2006 Share Posted November 3, 2006 (edited) Hi all,i wanna to Know What is the BEST way to solve equation of order 6?if you have equation : s^6+2S^5+S^4+8S^3+4S^2+15S+1562=0What will be the MOST easiest way to find its roots!! i know that you can use Bi-Section (Halfing Method) Any Other Ideas..N.B. I wanna to solve it manually without the use of PC or Calculator. Any ideas will be excellentRegards,Xtremee Edited November 8, 2006 by xtremee Link to comment Share on other sites More sharing options...
XP-is-a-CRAP Posted November 3, 2006 Share Posted November 3, 2006 (edited) if you have equation : s^6+2S^5+S^4+8S^3+4S^2+15S+1562=0WOW ... trying to remember about algebra ... What is the BEST way to solve equation of order 6?What will be the MOST easiest way to find its roots!!I wanna to solve it manually without the use of PC or Calculator.SORRY but it definitely IS the (a smart one) calculator or PC. Any Other Ideas..Well, since the highest exponent is 6 and divisible by 2, there is the risk that there is NO solution at all ... The number of solutions may be 0 ... 6 . There is a good method to solve equations of order 2 (possible result:no solution exists).There is a bad method for order 3 (exact solution for any equation)But there are NO (exact and universal) methods for order >=4 !!!What one can try is the brute-force method: test for solutions +1, -1, +2, -2, ...and reduce through polynom division ... works for "artificial" "school" equationswith low integer solutions only Otherwise, make a graph (preferably with a PC ), and then use aninexact method (tangent method, ...) to approach the solutions.BTW: If someone has the result (PC math prog) please post ... Edited November 3, 2006 by XP-is-a-CRAP Link to comment Share on other sites More sharing options...
IcemanND Posted November 3, 2006 Share Posted November 3, 2006 the equation posted is not possible.how about the constant being 62 instead of 1562?But of course the answer to everything is 42. Link to comment Share on other sites More sharing options...
tain Posted November 3, 2006 Share Posted November 3, 2006 I'm no good with math but I can tell you that it is spelled Mathematical not Mathimatical. Link to comment Share on other sites More sharing options...
LLXX Posted November 3, 2006 Share Posted November 3, 2006 There is a good method to solve equations of order 2 (possible result:no solution exists).There is a bad method for order 3 (exact solution for any equation)But there are NO (exact and universal) methods for order >=4 !!!Correction, a formula using conventional algebra exists for polynomials up to and including order 4 (quartic). It has been proved that no closed-form algebraic solution exists for polynomials of order 5 or higher.http://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Quartic_equation http://en.wikipedia.org/wiki/Quintic_equationBTW: If someone has the result (PC math prog) please post ...Here is Mathematica 4.1 attempting to solve your sextic equation. It cannot offer exact roots, as it does not have the capabilities to go beyond ordinary algebra. Below that is an approximation to each of the 6 roots (all complex numbers).http://img145.imageshack.us/img145/4372/equusuj6.png(Yes, "equus" is Latin for "horse".) Link to comment Share on other sites More sharing options...
gamehead200 Posted November 4, 2006 Share Posted November 4, 2006 Confirmed my my TI-89 and TI-83+ calculators. There are no solutions for that equation.However:d/dS(s^6+2S^5+S^4+8S^3+4S^2+15S+1562)= 6S^5+10S^4+4S^3+24S^2+8S+156S^5+10S^4+4S^3+24S^2+8S+15 = 0S = -2.18274Don't know if that will help... Link to comment Share on other sites More sharing options...
LLXX Posted November 4, 2006 Share Posted November 4, 2006 d/dS(s^6+2S^5+S^4+8S^3+4S^2+15S+1562)= 6S^5+10S^4+4S^3+24S^2+8S+156S^5+10S^4+4S^3+24S^2+8S+15 = 0S = -2.18274Fixed. Link to comment Share on other sites More sharing options...
ripken204 Posted November 6, 2006 Share Posted November 6, 2006 ya, just plug it into a calc and u should get the roots, although i havnt tried it yet. just graph and see where x=0 Link to comment Share on other sites More sharing options...
Zxian Posted November 6, 2006 Share Posted November 6, 2006 @gamehead - that'll give you where the original equation is flat (i.e. a maxima or minima). It unfortunately won't give you the zeros of the equation. Link to comment Share on other sites More sharing options...
gamehead200 Posted November 6, 2006 Share Posted November 6, 2006 I know. I just wanted to get in on the conversation! Link to comment Share on other sites More sharing options...
EchoNoise Posted November 6, 2006 Share Posted November 6, 2006 My head hurts Link to comment Share on other sites More sharing options...
RJM Posted November 7, 2006 Share Posted November 7, 2006 But S=2.9881167 gives you an error of only 0.000015 Link to comment Share on other sites More sharing options...
XP-is-a-CRAP Posted November 7, 2006 Share Posted November 7, 2006 @gamehead - that'll give you where the original equation is flat (i.e. a maxima or minima). It unfortunately won't give you the zeros of the equation.BUT I can check the "flat" points for minima, and if all of them are >0 then the fact is proven thatno usable (real numbers) solution exists.BTW: A flat point does NOT necessarily bring an extreme ... check "y=x^3" function.Anyone can generate and post the graph of the function on left side of the original equation, so we can see what we are trolling about ? Link to comment Share on other sites More sharing options...
LLXX Posted November 8, 2006 Share Posted November 8, 2006 The graph is very misleading. At different magnifications it looks very different.http://img218.imageshack.us/img218/6613/plot1bu7.pnghttp://img140.imageshack.us/img140/8462/plot2tg3.png Link to comment Share on other sites More sharing options...
xtremee Posted November 8, 2006 Author Share Posted November 8, 2006 (edited) @ all1st thanks for trying to help me2nd sorry for late in my replyi wanna an easy and quickly method that i can use to solve equation of order 6, 5 and 4 and i don't wanna the solution for 6 order equation that i post in the head of the topic. this equation for example only @gamehead200,Why you Diff. (d/ds) HOw it can help?Regards,Xtremee Edited November 8, 2006 by xtremee Link to comment Share on other sites More sharing options...
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