y-x-3-x-c-find-the-roots-0-c-2-3-3-

Question Number 140604 by ajfour last updated on 10/May/21

y = x^{3} - x - c; f i n d t h e r o o t s .

0 ⩽ c ⩽ \frac{2}{3 \sqrt{3}}

Answered by ajfour last updated on 11/May/21

Commented by ajfour last updated on 11/May/21

A E = 1 = E H

E P = x = t a n θ

E Q = x^{2} = {t a n}^{2} θ

H Q = x^{2} - 1

B H = c (n o t = 2 c a s i n i m a g e)

∠ B H Q = θ

\Rightarrow \frac{B H}{H Q} = t a n θ = \frac{c}{x^{2} - 1} = x

\Rightarrow x (x^{2} - 1) = c

l e t s f i n d o n e x i n t e r m s o f c .

A l s o \Rightarrow x^{2} - 1 = \frac{c}{x}

L e t c i r c u m r a d i u s ρ

O E = \frac{c}{2}

\Rightarrow ρ^{2} = \frac{c^{2}}{4} + x^{4}

a l s o ρ = \frac{c}{2} + x

\Rightarrow {(ρ - \frac{c}{2})}^{4} + \frac{c^{2}}{4} = ρ^{2}

l e t ρ - \frac{c}{2} = s; \frac{c}{2} = k

\Rightarrow s^{4} + k^{2} = s^{2} + 2 k s + k^{2}

\Rightarrow s^{3} - s - 2 k = 0

\Rightarrow s^{3} - s - c = 0

\dots \dots \dots

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