Question

If x square + 1/xsquare =66 ,find the value of x-1/x

Answer 1

${x}^{2} + \frac{1}{ {x}^{2} } = 66$

It can be written as,

${x}^{2} + \frac{1}{ {x}^{2} } = 64 + 2$

${x}^{2} + \frac{1}{ {x}^{2} } - 2 = 64$

${x }^{2} + \frac{1}{ {x}^{2} } - 2 \times x \times \frac{1}{x } = 64$

Now LHS is of the form

${(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab$

Therefore,

${(x - \frac{1}{x} )}^{2} = 64$

${( x- \frac{1}{x} )}^{2} = {8}^{2}$

$x - \frac{1}{x} = + 8 \: or \: - 8$

Answer 2

Step-by-step explanation:

$\bf \underline{Solution-}$

$\sf{ {x}^{2} + \frac{1}{ {x}^{2} } = 66}$

$\bf \underline{To find-}$

$\sf{the \: value \: of :\: x - \frac{1}{x} = \: ?}$

$\bf \underline{Solution-}$

$\sf {\bigg(x - \frac{1}{x} \bigg) ^{2} = {x}^{2} + \frac{1}{ {x}^{2} } - 2 \: \: \: \: \: \: [ \because \: (a - b {)}^{2} = {a}^{2} + {b}^{2} - 2ab ]} \\ \\ = 66 - 2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\rm{ [ \because {x}^{2} + \frac{1}{ {x}^{2}} = 66 \:(Given) ] }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:= 64$

$\sf{ \therefore \: \: \: \: \: \: x - \frac{1}{x} = \sqrt{64} } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = ± \: 8$

$\bf\underline{Hence,the \: value \: of : \: x - \frac{1}{x} \: is \: ± \: 8.}$