Question

If x = 5-2√6,
then find the value of x²+1/x²​

Answer 1

Given :-

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

To find :-

x²+1/x²

According to the question,

$\implies\sf{ \dfrac{1}{x} = \dfrac{1}{5 - 2 \sqrt{6} } \times \dfrac{5 + 2 \sqrt{6} }{5 + 2 \sqrt{6} } }$

$\implies \sf{ \frac{1}{x} = \frac{5 + 2 \sqrt{6} }{25 - 24} = 5 + 2 \sqrt{6 }}$

$\begin{gathered}\implies\sf{ x + \dfrac{1}{ x} = 5 - 2 \sqrt{6} + 5 + 2 \sqrt{6} } \\\end{gathered}$

$\implies \sf{x + \frac{1}{x}}$

Squaring on both the sides :

$\begin{gathered}\implies \sf{ {x}^{2} + \dfrac{1}{ {x}^{2} } = {10}^{2} } \\ \\ \implies \sf{ {x}^{2} + \frac{1}{ {x}^{2}} + 2 \times x \times \frac{1}{ x} = 100} \\ \\ \implies \sf{{x}^{2} + \frac{1}{ {x}^{2} } + 2= 100 } \\ \\ \implies \sf{ {x}^{2} + \frac{1}{ {x}^{2} } = 100 - 2 } \\ \\ { \underline{ \boxed{\implies {\sf{ {x}^{2} + \frac{1}{ {x}^{2} } = 98}}}}} \orange\bigstar\end{gathered}$

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Answer 2

Step-by-step explanation:

Given :

• If x = 5 + 2√6

To Find :

• the value of x²+1/x²

Solution :

x² = (5+2√6)²

= 5² + 2(5)(2√6) + (2√6)²

= 49 + 20√6

Then 1\x² = 1       

               ( 49 + 20√6)

Now, rationalising the denominator, we get,

 1               ×        49 - 20√6

49 + 20√6             49 - 20√6

  ______________________

=  49 - 20√6

Now, solve the equation

x² + 1/x² =  [ 49 + 20√6 ] + [ 49 - 20√6 ]

             =  49 + 20√6 + 49 - 20√6

= 49 + 49

             =  98

• Hence The x² + 1/x² is 98