Question

x=3+root2 then find the value of x2+1/x2

Answer 1

Given that

x= 3 + root2
So,

1/x = 1/3+root2
Rationalizing the denominator

So,
1/x =
$\frac{3 - \sqrt{2} }{(3 + \sqrt{2} )(3 - \sqrt{2} )} \\ \\ = \frac{3 - \sqrt{2} }{9 - 2} = \frac{3 - \sqrt{2} }{7}$
Now,

X +1/x =
$3 + \sqrt{2} + \frac{1}{7} \times (3 - \sqrt{2)} \\ = \frac{21 + 7 \sqrt{2} + 3 - \sqrt{2} }{7} \\ \\ = \frac{24 +6 \sqrt{2} }{7}$
Now,
Squaring both sides
${x}^{2} + \frac{1}{ {x}^{2} } + 2 = \frac{ {(24 - 6 \sqrt{2} )}^{2} }{49} \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = \frac{576 + 72 - 288 \sqrt{2} }{49} - 2 \\ \\ = \frac{648 - 288 \sqrt{2} - 98}{49} = \frac{550 - 28 \sqrt{2} }{49}$