Quintic equation by Hyperbolic Method

For quintic equations: if the roots are a,b,c,d,e.
Then, a = -(Rek + Re-k + Rel + Re-l)
a = -2R(cosh k + cosh l)
(cosh k + cosh l)5
= (cosh k + cosh l)2(cosh k + cosh l)3

= (cosh2 k + cosh2 l + 2 cosh k cosh l)(cosh k + cosh l)3

= (2cosh k cosh l)(cosh k + cosh l)3 + (cosh2 k + cosh2 l)(cosh k + cosh l)3

= (2cosh k cosh l)(cosh k + cosh l)3 + (cosh2 k + cosh2 l)(cosh k + cosh l)(cosh k + cosh l)2

= (2cosh k cosh l)(cosh k + cosh l)3 + (cosh3 k + cosh3 l + cosh k cosh2 l + cosh l cosh2 k)(cosh k + cosh l)2

= (2cosh k cosh l)(cosh k + cosh l)3 + (cosh3 k + cosh3 l + cosh k cosh l(cosh l + cosh k))(cosh k + cosh l)2

= (3cosh k cosh l)(cosh k + cosh l)3 + (cosh3 k + cosh3 l)(cosh k + cosh l)2

= (3cosh k cosh l)(cosh k + cosh l)3 + (cosh3 k + cosh3 l)(cosh k + cosh l)(cosh k + cosh l)

= (3cosh k cosh l)(cosh k + cosh l)3 + (cosh4 k + cosh4 l + cosh k cosh3 l + cosh3 k cosh l)(cosh k + cosh l)

= (3cosh k cosh l)(cosh k + cosh l)3 + (cosh4 k + cosh4 l + cosh k cosh l(cosh2 l + cosh2 k))(cosh k + cosh l)

= (3cosh k cosh l)(cosh k + cosh l)3 + (cosh4 k + cosh4 l)(cosh k + cosh l) + cosh k cosh l((cosh k + cosh l)2 - 2cosh k cosh l))(cosh k + cosh l)

= (3cosh k cosh l)(cosh k + cosh l)3 + (cosh4 k + cosh4 l)(cosh k + cosh l) + cosh k cosh l(cosh k + cosh l)3 - (2cosh2 k cosh2 l)(cosh k + cosh l)

= (4cosh k cosh l)(cosh k + cosh l)3 - (2cosh2 k cosh2 l)(cosh k + cosh l) + (cosh4 k + cosh4 l) (cosh k + cosh l)

= (4cosh k cosh l)(cosh k + cosh l)3 - (2cosh2 k cosh2 l)(cosh k + cosh l) + (cosh5 k + cosh5 l + cosh k cosh4 l + cosh4 l cosh l)

= (4cosh k cosh l)(cosh k + cosh l)3 - (2cosh2 k cosh2 l)(cosh k + cosh l) + (cosh5 k + cosh5 l) + (cosh k cosh l)(cosh3 k + cosh3 l)


(-2R)5(cosh k + cosh l)5 = (-2R)5(4cosh k cosh l)(cosh k + cosh l)3 - (-2R)5(2cosh2 k cosh2 l) (cosh k + cosh l) + (-2R)5(cosh5 k + cosh5 l) + (-2R)5(cosh k cosh l)(cosh3 k + cosh3 l)

(-2R)5(cosh k + cosh l)5 = (-2R)2(4cosh k cosh l)[-2R(cosh k + cosh l)]3 -(-2R)4(2cosh2 k cosh2 l) [-2R(cosh k + cosh l)] + (-2R)5(cosh5 k + cosh5 l) + (-2R)2(cosh k cosh l)[(-2R)3(cosh3 k + cosh3 l)]

(-2R)5(cosh k + cosh l)5 = (-2R)2(4cosh k cosh l)[-2R(cosh k + cosh l)]3 -(-2R)4(2cosh2 k cosh2 l) [-2R(cosh k + cosh l)] + (-2R)5(cosh5 k + cosh5 l) + (-2R)2(cosh k cosh l)[(-2R)3[(cosh k + cosh l)3 − 3(cosh k cosh l)(cosh k + cosh l)]

(-2R)5(cosh k + cosh l)5 = (-2R)2(5cosh k cosh l)[-2R(cosh k + cosh l)]3 -(-2R)4(5cosh2 k cosh2 l) [-2R(cosh k + cosh l)] + (-2R)5(cosh5 k + cosh5 l)

Write the quintic equation in the form x5 + cx3 + ex + f = 0

Then, (-2R)2(5cosh k cosh l)= - c
-(-2R)4(5cosh2 k cosh2 l) = - e
and (-2R)5(cosh5 k + cosh5 l) = -f

(-2R)2(cosh k cosh l)= -c/5
(-2R)4(cosh2 k cosh2 l) = e/5
The above two equalities state
[-c/5]2 = [e/5]

(-2R)2(cosh k cosh l)= -c/5
(-2R cosh l)= -c/(-2R ×5 cosh k)

(-2R)5(cosh5 k + cosh5 l) = -f
(-2R cosh k)5 + (-2R cosh l)5 = -f
(-2R cosh k)5 + (-c/(-2R ×5 cosh k))5 = -f
(-2R cosh k)10 + (-c/5)5 = -f(-2R cosh k)5
(-2R cosh k)10 + f(-2R cosh k)5 + (-c/5)5 = 0
(-2R cosh k)5 = [-f ± √{f2 - 4(-c/5)5}]/2
(-2R cosh k)= [[-f ± √{f2 - 4(-c/5)5}]/2]1/5


(-2R cosh l)= -c/(-2R ×5 cosh k)
= -c/5(-2R cosh k)
= -c/5[[-f ± √{f2 - 4(-c/5)5}]/2]1/5
= [[-f ∓ √{f2 - 4(-c/5)5}]/2]1/5
Hence the solution is ,
a = [[-f + √{f2 - 4(-c/5)5}]/2]1/5 + [[-f - √{f2 - 4(-c/5)5}]/2]1/5

Under the condition [-c/5]2 = [e/5]



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