Question

If x=3-2√2, find x^2+1/x^2​

Answer 1

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Answer 2

$x = 3 - 2 \sqrt{2}$

So,

$\frac{1}{x} = \frac{1}{3 - 2 \sqrt{2} } \\ = \frac{1}{3 - 2 \sqrt{2} } \times \frac{3 + 2 \sqrt{2} }{3 + 2 \sqrt{2} } \\ = \frac{3 + 2 \sqrt{2} }{ {3}^{2} - {(2 \sqrt{2} )}^{2} } \\ = \frac{3 + 2 \sqrt{2} }{9 - 8} = 3 + 2 \sqrt{2}$

Now,

${x}^{2} = (3 - 2 \sqrt{2} )^{2} = 9 + 8 - 12 \sqrt{2} \\ {x}^{2} = 17 - 12 \sqrt{2}$

And

$\frac{1}{ {x}^{2} } = (3 + 2 \sqrt{2} )^{2} = 9 + 8 + 12 \sqrt{2 } \\ \frac{1}{ {x}^{2} } = 17 + 12 \sqrt{2}$

therefore,

${x}^{2} + \frac{1}{ {x}^{2} } \\ = 17 - 12 \sqrt{2} + 17 + 12 \sqrt{2} \\ = 34$

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