Question

find the value of x+1/x​

Answer 1

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Answer 2

$\green\star\mathfrak\purple{\large{\underline{\underline{Answer:-}}}}$

$\bold \red{x = \frac{4}{3 - \sqrt{5} } }$

$\purple\star\mathfrak\green{\large{\underline{\underline{By \: rationalising:-}}}}$

$\implies \bold \red{x = \frac{4}{3 - \sqrt{5} } \times \frac{3 + \sqrt{5} }{3 + \sqrt{5} } }$

$\implies \bold \red{x = \frac{4(3 + \sqrt{5)} }{3 - \sqrt{5 } \times 3 + \sqrt{5} } }$

$\implies \bold \red{x = \frac{12 + 4 \sqrt{5} }{3 {}^{2} - (\sqrt{5}) {}^{2} } }$

$\implies \bold \red{x = \frac{12 + 4 \sqrt{5} }{9 - 5} }$

$\implies \bold \red{x = \frac{12 + 4 \sqrt{5} }{4} \: \: \: - (1)}$

$\green\star\bold\orange{\large{\underline{\underline{Now, \: \frac{1}{x} \: will \: be \: }}}}$

$\bold \pink{ \frac{1}{x} = \frac{4}{12 + 4 \sqrt{5} } }$

$\green\star\mathfrak\blue{\large{\underline{\underline{By \: rationalising:-}}}}$

$\implies \bold \pink{ \frac{1}{x} = \frac{4}{12 + 4 \sqrt{5} } \times \frac{12 - 4 \sqrt{5} }{12 - 4 \sqrt{5} } }$

$\implies \bold \pink{ \frac{1}{x} = \frac{4(12 - 4 \sqrt{5)} }{12 +4 \sqrt{5} \times 12 - 4 \sqrt{5} }{ } }$

$\implies \bold\pink { \frac{1}{x} = \frac{48 - 16 \sqrt{5} }{(12) {}^{2} - (4 \sqrt{5) {}^{2} } } }$

$\implies \bold \pink{ \frac{1}{x} = \frac{48 - 16 \sqrt{5} }{144 - 16 \times 5} }$

$\implies \bold \pink{ \frac{1}{x} = \frac{48 - 16 \sqrt{5} }{144 - 80} }$

$\implies \bold \pink{ \frac{1}{x} = \frac{48 - 16 \sqrt{5} }{64} \: \: - (2) }$

$\green\star\mathfrak\pink{\large{\underline{\underline{Now, According\:to\:the\: Question:-}}}}$

$\bold \green{x + \frac{1}{x} = \frac{12 + 4 \sqrt{5} }{4} + \frac{48 - 16 \sqrt{5} }{64} }$

$\implies \bold \green{ x + \frac{1}{x} = \frac{3 + \sqrt{5} }{1} + \frac{12 - 4 \sqrt{5} }{16} }$

$\implies \bold \green{x + \frac{1}{x} = 3 + \sqrt{5} + \frac{3 - \sqrt{5} }{4} }$

$\implies \bold \green{x + \frac{1}{x} = \frac{12 + 4 \sqrt{5} + 3 - \sqrt{5} }{4} }$

$\red \star\implies \bold \green{x + \frac{1}{x} = \frac{15 + 3 \sqrt{5} }{4} } \: \: \: \red\star$