x-4-x-2-cx-x-2-1-2-2-cx-1-4-let-x-2-1-2-t-t-2-1-4-2-c-2-t-1-2-now-let-t-2-1-4-z-z-2-c-2-2-2-c-4-z-1-4-z-2-c-2-2-p-p-2-c-4-4-2-c-8-p

Question Number 164341 by ajfour last updated on 16/Jan/22

x^{4} = x^{2} + c x

{(x^{2} - \frac{1}{2})}^{2} = c x + \frac{1}{4}

l e t x^{2} - \frac{1}{2} = t \Rightarrow

{(t^{2} - \frac{1}{4})}^{2} = c^{2} (t + \frac{1}{2})

n o w l e t t^{2} - \frac{1}{4} = z \Rightarrow

{(z^{2} - \frac{c^{2}}{2})}^{2} = c^{4} (z + \frac{1}{4})

z^{2} - \frac{c^{2}}{2} = p \Rightarrow {(p^{2} - \frac{c^{4}}{4})}^{2} = c^{8} (p + \frac{c^{2}}{2})

\Rightarrow p^{4} - \frac{c^{4} p^{2}}{2} - c^{8} p + \frac{c^{8}}{16} - \frac{c^{10}}{2} = 0

p^{4} - {A p}^{2} - B p - λ = 0

(p^{2} + s p + h) (p^{2} - s p - \frac{λ}{h}) = 0

h - \frac{λ}{h} = s^{2} - A

s (h + \frac{λ}{h}) = B

\frac{B^{2}}{s^{2}} - {(s^{2} - A)}^{2} = 4 λ

{(s^{2} - A)}^{3} + A {(s^{2} - A)}^{2} + 4 λ (s^{2} - A)

+ 4 λ A - B^{2} = 0

m^{3} + {A m}^{2} + 4 λ m + (4 λ A - B^{2}) = 0

l e t m = w - \frac{A}{3} \Rightarrow

w^{3} + (4 λ - \frac{A^{2}}{3}) w + \frac{2 A^{3}}{27} + \frac{8 λ A}{3} - B^{2} = 0

4 λ - \frac{A^{2}}{3} = 2 c^{10} - \frac{c^{8}}{4} - \frac{c^{8}}{12}

= 2 c^{8} (c^{2} - \frac{1}{6})

Commented by Tawa11 last updated on 16/Jan/22

Great sir

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