Question

If $x=9+4\sqrt{5}$ find $x^2+\frac{1}{x^2}$

Answer 1

Consider the given expression,

$x^2+\frac{1}{x^2}$ ……(1)

Put, the given value of $x=9+\sqrt{5}$ in equation 1^st, we get

${(9+4\sqrt{5})}^2+\frac{1}{{(9+4\sqrt{5})}^2}=81+16\times 5+2\times 4\sqrt{5}+\frac{1}{81+16\times 5+2\times 4\sqrt{5}}$

$=161+8\sqrt{5}+\frac{1}{161+8\sqrt{5}}=\frac{{(161+8\sqrt{5})}^2+1}{161+8\sqrt{5}}=\frac{25921+320+1288\sqrt{5}+1}{161+8\sqrt{5}}$

$=\frac{26242+1288\sqrt{5}}{161+8\sqrt{5}}$

Hence, this is the answer.