(((√(2x+1))+(√(x−1)))/( (√(2x+1))−(√(x−1)))) = (3/( (√(x+2))))


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Question Number 135486 by EDWIN88 last updated on 13/Mar/21

$\frac{\sqrt{2 x + 1} + \sqrt{x - 1}}{\sqrt{2 x + 1} - \sqrt{x - 1}} = \frac{3}{\sqrt{x + 2}}$
Commented by benjo_mathlover last updated on 13/Mar/21

$w r o n g$
Commented by benjo_mathlover last updated on 13/Mar/21

$\frac{\sqrt{2.1 + 1} + 0}{\sqrt{2.1 + 1} - 0} = \frac{1}{1} = 1 (L H S)$ $R H S \Rightarrow \frac{3}{\sqrt{3}} = \sqrt{3}$ $h o w \sqrt{3} = 1 ?$
Commented by Dwaipayan Shikari last updated on 13/Mar/21

$O o p p s! S o r r y$
	Answered by benjo_mathlover last updated on 13/Mar/21

	Answered by mr W last updated on 13/Mar/21

	$x ⩾ 1$ $\frac{{(\sqrt{2 x + 1} + \sqrt{x - 1})}^{2}}{x + 2} = \frac{3}{\sqrt{x + 2}}$ $2 x + 1 + x - 1 + 2 \sqrt{(2 x + 1) (x - 1)} = 3 \sqrt{x + 2}$ $2 \sqrt{(2 x + 1) (x - 1)} = 3 (\sqrt{x + 2} - x)$ $4 (2 x + 1) (x - 1) = 9 (x + 2 + x^{2} - 2 x \sqrt{x + 2})$ $x^{2} + 13 x + 22 = 18 x \sqrt{x + 2}$ $x^{4} - 298 x^{3} - 435 x^{2} + 572 x + 484 = 0$ $(x + 2) (x^{3} - 200 x^{2} + 165 x + 242) = 0$ $\Rightarrow x^{3} - 300 x^{2} + 165 x + 242 = 0$ $\Rightarrow x \approx 1.2175$
	Commented by MJS_new last updated on 13/Mar/21

	$you read my mind or I read yours ? anyway$ $you type faster than me . . .$
	Commented by mr W last updated on 13/Mar/21

	$h a h a, {i t}^{'} s t e l e p a t h y :)$
	Answered by MJS_new last updated on 13/Mar/21

	$\frac{{(\sqrt{2 x + 1} + \sqrt{x - 1})}^{2}}{x + 2} = \frac{3 \sqrt{x + 2}}{x + 2}$ $3 x + 2 \sqrt{(x - 1) (2 x + 1)} = 3 \sqrt{x + 2}$ $3 x = 3 \sqrt{x + 2} - 2 \sqrt{(x - 1) (2 x + 1)}$ $9 x^{2} = 9 (x + 2) + 4 (x - 1) (2 x + 1) - 12 \sqrt{(x - 1) (x + 2) (2 x + 1)}$ $14 + 5 x - x^{2} = 12 \sqrt{(x - 1) (x + 2) (2 x + 1)}$ ${(x^{2} - 5 x - 14)}^{2} = 144 (x - 1) (x + 2) (2 x + 1)$ $x^{4} - 298 x^{3} - 435 x^{2} + 572 x + 484 = 0$ $(x + 2) (x^{3} - 300 x^{2} + 165 x + 242) = 0$ $x = - 2 impossible$ $of the solutions of the 3^{rd} degree only one$ $solves the given equation :$ $x \approx 1.21750059$
	Answered by Dwaipayan Shikari last updated on 13/Mar/21

	$\frac{\sqrt{2 x + 1}}{\sqrt{x - 1}} = \frac{3 + \sqrt{x + 2}}{3 - \sqrt{x + 2}}$ $\Rightarrow \frac{\sqrt{2 x + 1}}{\sqrt{x - 1}} = \frac{(3 + \sqrt{x + 2})}{3 - \sqrt{x + 2}} \Rightarrow \frac{2 x + 1}{x - 1} = \frac{{(3 + \sqrt{x + 2})}^{2}}{{(3 - \sqrt{x + 2})}^{2}}$ $\Rightarrow \frac{3 x}{x + 2} = \frac{{(3 + \sqrt{x + 2})}^{2} + {(3 - \sqrt{x + 2})}^{2}}{{(3 + \sqrt{x + 2})}^{2} - {(3 - \sqrt{x + 2})}^{2}}$ $\Rightarrow \frac{3 x}{x + 2} = \frac{2 (9 + x + 2)}{12 \sqrt{x + 2}} \Rightarrow \frac{18 x}{\sqrt{x + 2}} = 11 + x \Rightarrow \frac{324 x^{2}}{x + 2} = 121 + x^{2} + 22 x$ $\Rightarrow 324 x^{2} = 121 x + x^{3} + 22 x^{2} + 242 + 2 x^{2} + 44 x$ $\Rightarrow x^{3} - 300 x^{2} + 165 x + 242 = 0$ $W h i c h g i v e s x \sim 1.2175$
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