Question

if a=3-2√2, find the value of a²-1/a².

Answer 1

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$a = 3 - 2 \sqrt{2} \\ \frac{1}{a} = \frac{1}{3 - 2 \sqrt{2} } = \frac{3 + 2 \sqrt{2} }{(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )} \\ \frac{1}{a} = \frac{3 + 2 \sqrt{2} }{9 - 8} = 3 + 2 \sqrt{2} \\ \\ now. \: a + \frac{1}{a} = 3 - 2 \sqrt{2} + 3 + 2 \sqrt{2} = 6 \\ \\ a - \frac{1}{a} = 3 - 2 \sqrt{2} - 3 - 2 \sqrt{2} = - 4 \sqrt{2} \\ \\ thus \: \: \: {a}^{2} - \frac{1}{ {a}^{2} } = (a + \frac{1}{a} )(a - \frac{1}{a} ) \\ = 6 \times ( - 4 \sqrt{2}) = - 24 \sqrt{2}$
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Answer 2

a=3−22a1=3−221=(3+22)(3−22)3+22a1=9−83+22=3+22now.a+a1=3−22+3+22=6a−a1=3−22−3−22=−42thusa2−a21=(a+a1)(a−a1)=6×(−42)=−242