Question

hiii!!!

$if \: {x}^{2} - \frac{1}{ {x}^{2} } = 66 \: then \: find \\ \: x - \frac{1}{x}$

content quality required with full explanation..

all the best :D

Answer 1

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


Square on both sides,


${( {x}^{2} - \frac{1}{ {x}^{2} } })^{2} = {66}^{2} \\ \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } - 2( {x}^{2} \times \frac{1}{ {x}^{2} } ) = 4356 \\ \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } - 2 = 4356 \\ \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = 4356 + 2 \\ \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = 4358$





Add 2( x² × 1 / x² ) on both sides,



$= > {x}^{4} + \frac{1}{ {x}^{4} } + 2( {x}^{2} \times \frac{1}{ {x}^{2} } ) = 4358 + 2( {x}^{2} \times \frac{1 }{ {x}^{2} } ) \\ \\ \\ = > {( {x}^{2} + \frac{1}{ {x}^{2} } })^{2} = 4358 + 2 \\ \\ \\ = > {( {x}^{2} + \frac{1 }{ {x}^{2} } )}^{2} = 4360 \\ \\ \\ = > {x}^{2} + \frac{ 1 }{ {x}^{2} } = \sqrt{4360}$





Add - 2( x × 1 / x ) on both sides,


${x}^{2} + \frac{1}{ {x}^{2} } - 2(x \times \frac{1}{x} ) = \sqrt{4360} - 2(x \times \frac{1}{x} ) \\ \\ \\ = > (x - \frac{1}{x} )^{2} = 60.03 \: - 2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{approx \: value \: of \: \sqrt{4360} = 60.03} \\ \\ \\ = > x - \frac{1}{x} = \sqrt{58.03} \\ \\ \\ = > x - \frac{1}{x} = \sqrt{58.03} \: \: \: or \: \: \: 7.6177$

Answer 2

thanks for the question,

We know,

a2 – b2 = (a – b)(a + b)

So, we get,

x2 - 1/x2 = 66

(x -1/x)( x+1/x)= 66

x- 1/x = 66 / {x +1/x}

x- 1/x = 66x / {x +1/}

(x2-1)/x = 66x /x2+1

(x2-1) (x2+1) = 66x2

Solving further,

x2 -1 = 66

x= square root of 67 = 8.06

So, x-1/x =8.18 - 1/8.18 =8.05

8.05 is the answer