((3x^2 −1)/x)+((5x)/(3x^2 −x−1))=((119)/(18))


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Question Number 198566 by tri26112004 last updated on 22/Oct/23

$\frac{3 x^{2} - 1}{x} + \frac{5 x}{3 x^{2} - x - 1} = \frac{119}{18}$
	Answered by Rasheed.Sindhi last updated on 22/Oct/23

	$\frac{3 x^{2} - 1}{x} + \frac{5 x}{3 x^{2} - x - 1} = \frac{119}{18}$ $\frac{3 x^{2} - 1}{x} - 1 + \frac{5 x}{3 x^{2} - x - 1} = \frac{119}{18} - 1$ $\frac{3 x^{2} - x - 1}{x} + 5 (\frac{x}{3 x^{2} - x - 1}) = \frac{101}{18}$ $\frac{3 x^{2} - x - 1}{x} = y :$ $y + 5 (\frac{1}{y}) = \frac{101}{18}$ $18 y^{2} - 101 y + 90$ $(2 y - 9) (9 y - 10) = 0$ $y = \frac{9}{2}, \frac{10}{9}$ $y = \frac{9}{2} :$ $\frac{3 x^{2} - x - 1}{x} = \frac{9}{2}$ $6 x^{2} - 11 x - 2 = 0$ $(x - 2) (6 x + 1) = 0$ $x = 2, x = - \frac{1}{6}$ $y = \frac{10}{9} :$ $\frac{3 x^{2} - x - 1}{x} = \frac{10}{9}$ $27 x^{2} - 19 x - 9 = 0$ $x = \frac{19 \pm \sqrt{361 + 972}}{54}$ $x = \frac{19 \pm \sqrt{1333}}{54}$
	Answered by Rasheed.Sindhi last updated on 22/Oct/23

	$A n O ther W ay$ $\frac{3 x^{2} - 1}{x} + \frac{5 x}{3 x^{2} - x - 1} = \frac{119}{18}$ $\frac{3 x^{2} - 1}{x} + 5 (\frac{1}{\frac{3 x^{2} - x - 1}{x}}) = \frac{119}{18}$ $\frac{3 x^{2} - 1}{x} + 5 (\frac{1}{\frac{3 x^{2} - 1}{x} - 1}) = \frac{119}{18}$ $y + \frac{5}{y - 1} = \frac{119}{18}; [\frac{3 x^{2} - 1}{x} = y]$ $18 y (y - 1) - 119 (y - 1) + 90 = 0$ $18 y^{2} - 18 y - 119 y + 119 + 90 = 0$ $18 y^{2} - 137 y + 209 = 0$ $(2 y - 11) (9 y - 19) = 0$ $y = \frac{11}{2}, \frac{19}{9}$ $y = \frac{11}{2} :$ $\frac{3 x^{2} - 1}{x} = \frac{11}{2}$ $6 x^{2} - 11 x - 2 = 0$ $(x - 2) (6 x + 1) = 0$ $x = 2, - \frac{1}{6}$ $y = \frac{19}{9} :$ $\frac{3 x^{2} - 1}{x} = \frac{19}{9}$ $27 x^{2} - 19 x - 9 = 0$ $x = \frac{19 \pm \sqrt{361 + 972}}{54}$ $x = \frac{19 \pm \sqrt{1333}}{54}$
	Answered by Rasheed.Sindhi last updated on 22/Oct/23

	$\frac{3 x^{2} - 1}{x} + \frac{5 x}{3 x^{2} - x - 1} = \frac{119}{18}$ $\frac{(3 x^{2} - 1) \div x}{x \div x} + \frac{5 x \div x}{(3 x^{2} - x - 1) \div x} = \frac{119}{18}$ $3 x - \frac{1}{x} + \frac{5}{3 x - 1 - \frac{1}{x}} = \frac{119}{18}$ $3 x - \frac{1}{x} = y$ $y + \frac{5}{y - 1} = \frac{119}{18}$ $18 y^{2} - 137 y + 29 = 0$ $F o r t h e r e m a i n i n g s e e m y o t h e r$ $a n s w e r .$
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