1.

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

Answer»

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 } \:\green\bigstar}

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

\implies \sf{x + \frac{1}{x} = 10 \: \pink\bigstar }

SQUARING on both the SIDES :

\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


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