If x + ((49)/(x + 48)) = − 34 find (2x + 83)^3 + (1/((2x + 83)^3 ))


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Question Number 209240 by Jubr last updated on 05/Jul/24

$I f x + \frac{49}{x + 48} = - 34$ $f i n d {(2 x + 83)}^{3} + \frac{1}{{(2 x + 83)}^{3}}$
	Answered by som(math1967) last updated on 05/Jul/24

	$x + 48 + \frac{49}{x + 48} = - 34 + 48$ $\Rightarrow a + \frac{49}{a} = 14 [l e t x + 48 = a]$ $\Rightarrow a^{2} - 14 a + 49 = 0$ $\Rightarrow {(a - 7)}^{2} = 0$ $∴ a = 7$ $\Rightarrow x + 48 = 7$ $\Rightarrow x = - 41$ $\Rightarrow 2 x = - 82$ $\Rightarrow 2 x + 83 = 1$ $∴ (2 x + 83) + \frac{1}{(2 x + 83)} = 1 + 1$ $y + \frac{1}{y} = 2 [l e t (2 x + 83) = y]$ ${(y + \frac{1}{y})}^{3} = 8$ $\Rightarrow y^{3} + \frac{1}{y^{3}} + 3 . y . \frac{1}{y} (y + \frac{1}{y}) = 8$ $\Rightarrow y^{3} + \frac{1}{y^{3}} = 8 - 3 \times 2 = 2$ $∴ {(2 x + 83)}^{3} + \frac{1}{{(2 x + 83)}^{3}} = 2$
	Commented by Frix last updated on 05/Jul/24

	$Yes!$ $But$ $2 x + 83 = 1 \Rightarrow {(2 x + 83)}^{n} + \frac{1}{{(2 x + 83)}^{n}} = 2$ $without any further calculations .$
	Commented by Spillover last updated on 05/Jul/24

	$p e r f e c t$
	Commented by Tawa11 last updated on 05/Jul/24

	$Thanks sir . I appreciate .$
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