v-2b-3cp-p-3-bp-2-3v-2-2bpv-3cp-b-1-bp-2-cp-3-gt-0-b-c-R-b-lt-0-Any-non-zero-real-value-of-p-in-terms-of-b-c-obeying-above-condition-

Question Number 80881 by ajfour last updated on 07/Feb/20

v = - \frac{(2 b + 3 c p) p}{3 + {b p}^{2}}

\frac{3 v^{2} + 2 b p v + 3 c p + b}{1 + {b p}^{2} + {c p}^{3}} > 0

b, c \in R, b < 0

A n y n o n - z e r o r e a l v a l u e o f p

i n t e r m s o f b, c o b e y i n g a b o v e

c o n d i t i o n ?

Answered by ajfour last updated on 07/Feb/20

l e t p = - \frac{b}{c}

v = \frac{- b^{2}}{c (3 + \frac{b^{3}}{c^{2}})} = - \frac{b^{2} c}{3 c^{2} + b^{3}}

S = {(\frac{b^{2} c}{3 c^{2} + b^{3}})}^{2} + \frac{2 b^{2}}{c} (\frac{b^{2} c}{3 c^{2} + b^{3}}) - 2 b

l e t m e t e s t i t

x^{3} - 49 x + 120 = 0

l e t x = \frac{u + v}{1 + p u}

u^{3} + S u + \frac{v^{3} + b v + c}{1 + {b p}^{2} + {c p}^{3}} = 0

\Rightarrow u^{3} + S u + T = 0

D = \frac{T^{2}}{4} + \frac{S^{3}}{27}

u = {- \frac{T}{2} + \sqrt{D}}^{1 / 3} + {- \frac{T}{2} - \sqrt{D}}^{1 / 3}

\dots \dots . .

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