Question

If a-1/a=3, find the value of a^2-1/a^2

Answer 1

$a - \frac{1}{a} = 3 \: \: \: \: \: \: \: \: \: \: \: \: \: \: ... \: (i) \\ {(a - \frac{1}{a} )}^{2} = {(3)}^{2} \\ {a}^{2} + \frac{1}{ {a}^{2} } - 2(a)( \frac{1}{a} ) = 9 \\ {a}^{2} + \frac{1}{ {a}^{2} } - 2 = 9 \\ {a}^{2} + \frac{1}{ {a}^{2} } = 11 \\ {(a + \frac{1}{a} )}^{2} - 2(a)( \frac{1}{a} ) = 11 \\ {(a + \frac{1}{a} )}^{2} - 2 = 11 \\ {(a + \frac{1}{a} )}^{2} = 13 \\ a + \frac{1}{a} = \sqrt{13} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ... \: (ii)\\ \\ now \\ \\ \: \: \: \: \: {a}^{2} - \frac{1}{ {a}^{2} } \\ = (a - \frac{1}{a})(a + \frac{1}{a} ) \\ = (3)( \sqrt{13} ) \: \: \: \: \: \: \: \: \: \: \: ... \: (from \: i \: and \: ii) \\ = 3 \sqrt{13}$