Question

If x - 1/2x =3, Find the value of x2+1/4x2


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Answer 1

Answer:

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Answer 2

⇨GIVEN:-

$\tt ↠ \red{2x = 3 + \sqrt{7} }$

⇨FIND:-

$\tt ↠ \blue{ {4x}^{2} + \frac{1}{ {x}^{2} } }$

⇨SOLUTION:-

$\tt⤳ \green{2x = 3 + \sqrt{7} }.....(i)$

$\tt➾ \frac{1}{x} = \frac{2}{3 + \sqrt{7} } \\ \tt \gray{❅ rationalise \: the \: denominator} \\ \tt ➾ \frac{2}{3 + \sqrt{7} } \times \frac{3 - \sqrt{7} }{3 - \sqrt{7} } \\۞\tt by \: using \: identity \: (a + b)(a - b) = {a}^{2} - {b}^{2} \: we \: have\\ \tt \frac{2(3 - \sqrt{7} )}{ {3}^{2} - ( \sqrt{7} {)}^{2} } = \frac{2(3 - \sqrt{7} )}{ 9 - 7 } \\ \tt➾ \frac{ \cancel2(3 - \sqrt{7} )}{ \cancel2} \\ \tt ➾ \pink{ \frac{1}{x} = 3 - \sqrt{7}} .....(ii)$

adding eq(i) and (ii) we have,

$\tt⪼2x + \frac{1}{x} = 3 \cancel{+ \sqrt{7} } + 3 \cancel{- \sqrt{7} }$

$\tt⪼2x + \frac{1}{x} = 3 + 3 \\ \tt⪼2x + \frac{1}{x} = 6$

now, squaring both sides we have,

$\tt \implies {(2x + \frac{1}{x}) }^{2} = {6}^{2}$

$\tt ✼by \: using {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \: we \: have$

$\tt \implies {(2x)}^{2} + \frac{(1 {)}^{2} }{ {x}^{2} } + 2 \times 2 \cancel x \times \frac{1}{ \cancel x } = 36$

$\tt \implies 4 {x}^{2} + \frac{1}{ {x}^{2} } + 4 = 36$

$\tt \implies 4 {x}^{2} + \frac{1}{ {x}^{2} } = 36 - 4$

$\tt \implies 4 {x}^{2} + \frac{1}{ {x}^{2} } = 32$

$\tt Hence, \huge\boxed{ \bold{ 4 {x}^{2} + \frac{1}{ {x}^{2} } = 32}}$