Question

If x= 3+2√2, find the value of x^2+1÷x^2​

Answer 1

Answer:

$x = 3 + 2 \sqrt{2} \\ \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = (3 + 2 \sqrt{2}) ^{2} + \frac{1}{(3 + 2 \sqrt{2})^{2} } \\ \\ = 9 + 8 + 12\sqrt{2} + \frac{1}{(9 + 8 + 12 \sqrt{2}) } \\ \\ = 17 + 12 \sqrt{2} + \frac{1}{17 + 12 \sqrt{2} } \\ \\ \\ = \frac{(17 + 12 \sqrt{2} )^{2} + 1 }{17 + 12 \sqrt{2} } \\ \\ = \frac{ {17}^{2} + (12 \sqrt{2} )^{2} + 2 \times 17 \times 12 \sqrt{2} + 1 }{17 + 12 \sqrt{2} } \\ \\ \\ = \frac{289 + 288 + 408 \sqrt{2} + 1 }{17 + 12\sqrt{2} } \\ \\ \\ \\ = \frac{578 + 408 \sqrt{2} }{17 + 12 \sqrt{2} } \\ \\ \\ \\ = \frac{34(17 + 12 \sqrt{2}) }{17 + 12 \sqrt{2} } \\ \\ \\ = 34$

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