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Question

If $x = 9 + 4 \sqrt{5}$ , find the value of $x^{2} + \frac{1}{x^{2}}$ .

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Solution

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We have,
$x = 9 + 4 \sqrt{5}$ .
Then,
$\frac{1}{x}$
$= \frac{9 - 4 \sqrt{5}}{81 - 80}$
$= 9 - 4 \sqrt{5}$ .
Now,
$x^{2} + \frac{1}{x^{2}}$
$= {(x + \frac{1}{x})}^{2} - 2$
$= 18^{2} - 2$
$= 324 - 2$
$= 322$ .

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