If $x + 1 / x = 3$ , find $x^{2} + 1 / x^{2}$

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Solution

$x + \frac{1}{x} = 3 \Rightarrow {(x + \frac{1}{x})}^{2} = 9$
$\Rightarrow x^{2} + \frac{1}{x^{2}} + 2 \times x \times \frac{1}{x} = 9$
$\Rightarrow x^{2} + \frac{1}{x^{2}} = 9 - 2$
$\Rightarrow x^{2} + \frac{1}{x^{2}} = 7$

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