Question

if x=5-2√6, find
(i) x2-1/x2
(ii) √x-1/√x​

Answer 1

Answer:

$x = 5 - 2 \sqrt{6}$

$(1) \: \: \: {x}^{2} - \frac{1}{ {x}^{2} }$

$= {(5 - 2 \sqrt{6)} }^{2} - \frac{1}{ {(5 - 2 \sqrt{6} )}^{2} }$

$= 25 + 24 - 20 \sqrt{6} + \frac{1}{25 + 24 - 2 \sqrt{6} }$

$= 49 - 20 \sqrt{6} + \frac{1}{49 - 20 \sqrt{6} }$

$= \frac{49(49 - 20 \sqrt{6}) - 20 \sqrt{6} (49 - 20 \sqrt{6} ) }{49 - 20 \sqrt{6} }$

$= \frac{2401 - 980 \sqrt{2} - 980 \sqrt{2} -2400 }{49 - 20 \sqrt{6} }$

$= \frac{1 - 1960 \sqrt{2} }{49 - 20 \sqrt{6} }$

$(2) \sqrt{x} - \frac{1}{ \sqrt{x} }$

$\sqrt{5 - 2 \sqrt{6} } - \frac{1}{ \sqrt{5 - 2\sqrt{6} } }$

$= \frac{5 - 2 \sqrt{6} - 1 }{ \sqrt{5 - 2 \sqrt{6} } }$

$= \frac{4 - 2 \sqrt{6} }{ \sqrt{5 - 2 \sqrt{6} } }$