reposting this:
x^4 +(1−2a)x^2 −2ax+1=0
solve for a>0∧x>0
a^5 −((23)/8)x^4 −((49)/4)x^3 +((27)/8)x^2 +3x+(9/8)=0
⇒ 3 real solutions a_1 <a_2 <a_3
we cannot solve exactly (?) so it′s not
possible to get all the below solutions
exactly, only rough approximations
for the following α,β,γ,δ ∈R
the possible natures of the zeros:
(1) a<a_1 ∨a_2 <a<a_3 ⇒ x=α∨x=β∨x=γ±δi
(2) a>a_3 ⇒ x=α∨x=β∨x=γ∨x=δ
(3) a_1 <a<a_2 ⇒ x=α±βi∨x=γ±δi
(4) a=a_3 ⇒ x=α=β∨x=γ∨x=δ
(5) a=a_1 ∨a=a_2 ⇒ x=α=β∨x=γ±δi
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