x-3-x-c-0-solve-for-x-even-if-0-lt-c-lt-2-3-3-

Question Number 129687 by ajfour last updated on 17/Jan/21

x^{3} - x - c = 0

(s o l v e f o r x e v e n i f 0 < c < \frac{2}{3 \sqrt{3}})

Answered by ajfour last updated on 18/Jan/21

J u s t n o w I d e v e l o p e d a n e w

a p p r o x i m a t e f o r m u l a :

x = - {(\frac{c}{2})}^{1 / 3} + \sqrt{5 {(\frac{c}{2})}^{2 / 3} + \frac{4}{3}}

e x a m p l e :

y = (X - 7) (X + 2) (X + 5)

= X^{3} - 39 X - 70

y = 0 \Rightarrow X^{3} - 39 X - 70 = 0

l e t x = \frac{X}{\sqrt{39}}

\Rightarrow x^{3} - x - \frac{70}{39 \sqrt{39}} = 0

h e r e c = \frac{70}{39 \sqrt{39}}

x = - \frac{{(35)}^{1 / 3}}{\sqrt{39}} + \sqrt{5 \times \frac{{(35)}^{2 / 3}}{39} + \frac{4}{3}}

X = \sqrt{39} x

X = - {(35)}^{1 / 3} + \sqrt{5 {(35)}^{2 / 3} + \frac{4 \times 39}{3}}

X = 7.00022

(E u r e k a!)

Commented by Dwaipayan Shikari last updated on 18/Jan/21

H o w d i d y o u d e r i v e i t s i r ?