Question

find the answer
$if \: \: x \: = \: 3 + 2 \sqrt{2} \: \: then \: find \: the \: value \: of \: \: {x}^{2} + \frac{1}{ {x}^{2} }$
please​

Answer 1

Question :

If x = 3 + 2√2 , then find the value of x² + 1/x²

Solution :

If x = 3 + 2√2 then for 1/x we get

$\looparrowright \frac{1}{3 + 2 \sqrt{2} }$

By rationalising the denominator we get

$\looparrowright \frac{1(3 - 2 \sqrt{2} )}{(3 + 2 \sqrt{2})(3 - 2 \sqrt{2} )}$

By applying formula (a + b)(a - b) = a² - b² we get

$\looparrowright \frac{3 - 2 \sqrt{2} }{ {(3)}^{2} - {(2 \sqrt{2} )}^{2} }$

$\looparrowright \frac{3 - 2 \sqrt{2} }{9- 8}$

$\looparrowright \frac{3 - 2 \sqrt{2} }{1}$

$\looparrowright \frac{1}{x} \rarr3 - 2 \sqrt{2}$

Now as per the given we now need to simplify accordingly

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

By putting the values

$\sf \leadsto {(3 + 2 \sqrt{2} )}^{2} + {(3 - 2 \sqrt{2} )}^{2}$

By applying formula :

• (a + b)² = a² + 2ab + b² [For the L.H.S]

• (a - b)² = a² - 2ab + b² [For the R.H.S]

$\sf \leadsto{(3)}^{2} + 2 \times 3 \times 2 \sqrt{2} + {(2 \sqrt{2} )}^{2} + {(3)}^{2} - 2 \times 3 \times 2 \sqrt{2} + {(2 \sqrt{2}) }^{2}$

$\sf \leadsto{(3)}^{2} + \cancel{2 \times 3 \times 2 \sqrt{2}} + {(2 \sqrt{2} )}^{2} + {(3)}^{2} - \cancel{ 2 \times 3 \times 2 \sqrt{2} } + {(2 \sqrt{2}) }^{2}$

$\sf \leadsto{(3)}^{2} + {(2 \sqrt{2} )}^{2} + {(3)}^{2} + {(2 \sqrt{2}) }^{2}$

$\sf \leadsto9 + 8 + 9 + 8$

$\sf \leadsto34$

So the value of x² + 1/x² is 34