Question

if x=3+2√2 then xsquare+1/xsquare

Answer 1

Hey friend,
Here is the answer you were looking for:
$x = 3 + 2 \sqrt{2} \\ \\ \frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} }$

On rationalizing the denominator we get :

$\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} }$

Using the identity :

$(x + y)(x - y) = {x}^{2} - {y}^{2}$

$\frac{1}{x} = \frac{3 + 2 \sqrt{2} }{ {(3)}^{2} - {(2 \sqrt{2}) }^{2} } \\ \\ \frac{1}{x} = \frac{3 + 2 \sqrt{2} }{9 - 8} \\ \\ \frac{1}{x} = 3 + 2 \sqrt{2} \\ \\ x + \frac{1}{x} = (3 - 2 \sqrt{2} ) + (3 + 2 \sqrt{2} ) \\ \\ x + \frac{1}{x} = 3 - 2 \sqrt{2} + 3 + 2 \sqrt{2} \\ \\ x + \frac{1}{x} = 6$

On squaring both the sides we get,

${(x + \frac{1}{x}) }^{2} = {(6)}^{2} \\ \\ {x}^{2} + {( \frac{1}{x} )}^{2} + 2 \times x \times \frac{1}{x} = 36 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 36 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 36 - 2 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 34$

Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
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