I started with a quick graph which strongly hinted there would be no solution. There are 4 real solutions for a>120
https://www.desmos.com/calculator/mmsdjd8elb
Call the roots (c-3d), (c-d), (c+d), (c+2d)
Expand the polynomial
(x-(c-3d))(x-(c-d))(x-(c+d))(x-(c+3d))
Divide the original polynomial by 16 and equate the terms gives the system [numbered by power of x]:
[3] -4c = -a/16
[2] 6c^2-10d^2 = (2a+17)/16
[1] -2c(2c^2-10d^2) = -a/16
[0] c^4-10c^2d^2+9d^4 = 0
A system with 4 equations for 3 unknowns.
From [3] a=64c and we can sub this in to the others to get a new system:
[2'] 6c^2-10d^2 = (128c+17)/16
[1'] (2c^2-10d^2) = 2
[0] c^4-10c^2d^2+9d^4 = 0
This can be graphed (c,d)=(x,y) to see that there are no solutions.
https://www.desmos.com/calculator/qahbjh8efg
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Posted by Jer
on 2024-08-01 15:35:00 |
No solution
