Question

If x= 5+2√6, find x² + 1/x²

Answer 1

If x = 5 + 2√6
Then x² = (5+2√6)² = 5² + 2(5)(2√6) + (2√6)² = 49 + 20√6
Then 1\x² =      1       
               ( 49 + 20√6)
Now, rationalising the denominator, we get,
 1               ×        49 - 20√6
49 + 20√6             49 - 20√6
  
=  49 - 20√6

Now, solve the equation
x² + 1/x² =  [ 49 + 20√6 ] + [ 49 - 20√6 ]
             =  49 + 20√6 + 49 - 20√6
             =  98 Is the answer

Answer 2

given

$x = 5 + 2 \sqrt{6}$

we can write

$\frac{1}{x} = \frac{1}{5 + 2 \sqrt{6} }$

rationalizing denominator we get

$\frac{1}{x} = \frac{5 - 2 \sqrt{6} }{25 - 24} \\ \\ \frac{1}{x} = 5 - 2 \sqrt{6}$

Now

$x + \frac{1}{x} = 10$

squaring both sides we get

$( {x + \frac{1}{x} })^{2} = 100 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 100 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 98$