find-the-inverse-of-y-x-4-2x-1-

Question Number 124619 by arkanmath7@gmail.com last updated on 04/Dec/20

f i n d t h e i n v e r s e o f y = x^{4} - 2 x + 1

Commented by MJS_new last updated on 04/Dec/20

x^{4} - 2 x - (y + 1) = 0

x = ?

can only be solved using the complete path

of Ferrari . I found no easier path; if you have

a solution, please share it

Commented by arkanmath7@gmail.com last updated on 04/Dec/20

A c t u a l l y I l o o k f o r t h e a n s w e r

Answered by ajfour last updated on 04/Dec/20

y = (x^{2} - m x + p) (x^{2} + m x + q)

p + q = m^{2}

m (p - q) = - 2

p q = 1

l e t p + q = λ

λ (λ^{2} - 4) = 4

λ^{3} - 4 λ - 4 = 0

D = 4 - \frac{64}{27} > 0

s o λ f r o m {C a r d a n o}^{'} s .

λ \approx 2.38298

f o r p a n d q

z^{2} - λ z + 1 = 0

p, q = z = \frac{λ}{2} \pm \sqrt{\frac{λ^{2}}{4} - 1}

= 0.54369, 1.83929

l e t p > q \Rightarrow p = 1.83929, q = 0.54369

m = - \frac{2}{p - q} = - 1.54368632

x^{2} + m x + q = 0

x^{2} - 1.54368632 x + 0.54369 = 0

x = 0.54369, 1

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