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        riddles >> easy >> A Quartic Evaluation
        
(Message started by: K Sengupta on Dec 6th, 2005, 12:21am)
Title: A Quartic Evaluation
Post by K Sengupta on Dec 6th, 2005, 12:21am
Given that H is a real number, determine the number of real roots of the quartic  equation:

      x4+(1-2H)x2 + H2 - 1 = 0;  

Title: Re: A Quartic Evaluation
Post by towr on Dec 6th, 2005, 4:01am
[hideb]
y = x2

y2+(1-2H)y + H2 - 1 = 0;    

y1,2 = [-(1-2H) +/- sqrt((1-2H)2 -  4(H2 - 1))]/2 = [2H-1 +/- sqrt(5 - 4 H)]/2

H = 5/4:
 y = (2H-1)/2 > 0  ==> 2 real solutions
H < 5/4:
 y1,2 = [2H-1 +/- sqrt(5 - 4 H)]/2
 H > 1 ==> 4 real solutions
 H = 1 ==> 3 real solutions
 H < 1 & H > -1 ==> 2 real solutions
 H = -1 ==> 1 real solution
 H < -1 ==> 0 real solutions
H > 5/4:
 0 real solutions
[/hideb]
Title: Re: A Quartic Evaluation
Post by K Sengupta on Dec 11th, 2005, 10:25pm
If. in addition it is given that K is  also a real number, determine the number of real roots corresponding to each of the undernoted quartic  equation:

(i) x4+(1-2K)x2 + H2 - 1 = 0;

 




Title: Re: A Quartic Evaluation
Post by towr on Dec 12th, 2005, 4:48am
Aren't those two basicly the same? Just H and K are exchanged.
Title: Re: A Quartic Evaluation
Post by K Sengupta on Dec 15th, 2005, 9:15pm
Of course, you are right. Problem (ii) was oversight on my part, and any inconvenience caused due to the foregoing is sincerely regretted.

All the texts inclusive of  Problem (ii) stands expunged with immediate effect with this updation.

I now append hereunder, an extension of   tenets inclusive of the  Problem entitled "A Quartic Evaluation":

Given that H,K and L are real numbers, determine the total number of real roots of the undernoted Quartic equation:

L x4+(1-2K)x2 + H2 - 1 = 0;


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