x^3 −x−c=0 (solve for x even if 0<c<(2/(3(√3))))


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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 byDwaipayan Shikari last updated on 18/Jan/21

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