Question

if x=5-2√6 then find the value of x 2+1 /x 2

Answer 1

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Answer 2

Hey mate!

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Given :

$x = 5 - 2 \sqrt{6}$

To find :

$x {}^{2} + \frac{1}{x {}^{2} }$

Solution :

$x = 5 - 2 \sqrt{6} \\ \\ \frac{1}{x} = \frac{1}{5 - 2 \sqrt{6} } \times \frac{5 + 2 \sqrt{6} }{5 + 2 \sqrt{6} } \\ \\ \frac{1}{x} = \frac{5 + 2 \sqrt{6} }{(5) {}^{2} - (2 \sqrt{6}) {}^{2} } \\ \\ \frac{1}{x} = \frac{5 + 2 \sqrt{6} }{25 - 24} \\ \\ \frac{1}{x} = 5 + 2 \sqrt{6}$

Now,

$x + \frac{1}{x} = 5 - \cancel {2 \sqrt{6}} + 5 + \cancel {2 \sqrt{6}} \\ \\ x + \frac{1}{x} = 5 + 5 \\ \\ x + \frac{1}{x} = 10$

And, on squaring both sides ;

$(x + \frac{1}{x} ) {}^{2} = (10) {}^{2} \\ \\ x {}^{2} + \frac{1}{x {}^{2} } + 2 = 100 \\ \\ x {}^{2} + \frac{1}{x {}^{2} } = 100 - 2 \\ \\ x {}^{2} + \frac{1}{x {}^{2} } = 98$


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Thanks for the question!

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