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Topic: A Quartic Evaluation (Read 311 times) |
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K Sengupta
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A Quartic Evaluation
« on: Dec 6th, 2005, 12:21am » |
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Given that H is a real number, determine the number of real roots of the quartic equation:
x4+(1-2H)x2 + H2 - 1 = 0;
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towr
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Re: A Quartic Evaluation
« Reply #1 on: Dec 6th, 2005, 4:01am » |
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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
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| « Last Edit: Dec 6th, 2005, 4:24am by towr » |
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K Sengupta
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Re: A Quartic Evaluation
« Reply #2 on: Dec 11th, 2005, 10:25pm » |
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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;
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| « Last Edit: Dec 15th, 2005, 9:16pm by K Sengupta » |
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towr
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Re: A Quartic Evaluation
« Reply #3 on: Dec 12th, 2005, 4:48am » |
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Aren't those two basicly the same? Just H and K are exchanged.
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K Sengupta
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Re: A Quartic Evaluation
« Reply #4 on: Dec 15th, 2005, 9:15pm » |
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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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| « Last Edit: Dec 15th, 2005, 9:17pm by K Sengupta » |
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