Question

if x=9+4√5 then find the value of x^2-1/x^2​

Answer 1

★ x=9+4√5.

=> x²=(9+4√5)²

=>x²=81+80+72√5.

=>x²=161+72√5.

★x²=161+72√5.

=>1/x²=1/161+72√5.

=>1/x²=(161-72√5)/(161+72√5)(161-72√5)

=>1/x²=161-72√5.

★x²+1/x²

=161+72√5+161-72√5

=322.

Answer 2

$x = 9 + 4 \sqrt{5} \\ \frac{1}{x} = 9 - 4 \sqrt{5} (by \: rationalizing \: denominator) \\ \\ x + \frac{1}{x} = 18 \\ \\ x - \frac{1}{x} = 8 \sqrt{5} \\ \\ {x}^{2} - \frac{1}{ {x}^{2} } = (x + \frac{1}{x} )(x - \frac{1}{x} ) \\ = 18 \times 8 \sqrt{5} \\ = 144 \sqrt{5}$