Math, asked by smitaranimohanty7551, 5 hours ago

3. If x = 5 - 2√6, find the value of : (i) x+1/x (ii) x2 + 1/x2 √5​

Answers

Answered by 12thpáìn
3

Given

  • x = 5 - 2√6

To Find

  1. x+1/x
  2. x² + 1/x²

Solution

 \sf  \:  \: \implies x -  \dfrac{1}{x}

 \sf  \:  \: \implies {5 - 2 \sqrt{6} -  \dfrac{1}{5 - 2 \sqrt{6} }  }

 \sf  \:  \: \implies {5 - 2 \sqrt{6} -  \dfrac{1}{5 - 2 \sqrt{6} }  \times  \dfrac{5 + 2 \sqrt{6} }{5  +  2 \sqrt{6} }  }

 \sf  \:  \: \implies {5 - 2 \sqrt{6} -  \dfrac{5 + 2 \sqrt{6} }{ {5}^{2}  -( 2 \sqrt{6}    )^{2}} }

 \sf  \:  \: \implies {5 - 2 \sqrt{6} -  \dfrac{5 + 2 \sqrt{6} }{ 25 -   24} }

 \sf  \:  \: \implies {5 - 2 \sqrt{6}  - 5 - 2 \sqrt{6} }

 \sf  \:  \: \implies {4 \sqrt{6}  }

___________________

 \sf  \:  \:  \:  \implies{{x}^{2}  +  \dfrac{1}{ {x}^{2} } }

 \sf  \:  \:  \:  \implies{{(5 - 2 \sqrt{6}) }^{2}  +  \dfrac{1}{ {(5 - 2 \sqrt{6}) }^{2} } }

 \sf  \:  \:  \:  \implies{{( {5}^{2}  +  24 - 20 \sqrt{6}  )}  +  \dfrac{1}{ {( {5}^{2}  +  24 - 20 \sqrt{6}  )} } }

 \sf  \:  \:  \:  \implies{{ 25  +  24 - 20 \sqrt{6}  }  +  \dfrac{1}{ 25 +  24 - 20 \sqrt{6}  } }

 \sf  \:  \:  \:  \implies{{ 49 - 20 \sqrt{6}  }  +  \dfrac{1}{ 49 - 20 \sqrt{6}  } }

\sf  \:  \:  \:  \implies{{ 49 - 20 \sqrt{6}  }  +  \dfrac{1}{ 49 - 20 \sqrt{6}  } \times  \dfrac{49 + 20 \sqrt{6} }{49 + 20 \sqrt{6} }  }

\sf  \:  \:  \:  \implies{{ 49 - 20 \sqrt{6}  }  +  \dfrac{49 + 20 \sqrt{6} }{  {49}^{2}  - (20 \sqrt{6}  ) ^{2}  } }

\sf  \:  \:  \:  \implies{{ 49 - 20 \sqrt{6}  }  +  \dfrac{49 + 20 \sqrt{6} }{  2401  - 400  \times 6} }

\sf  \:  \:  \:  \implies{{ 49 - 20 \sqrt{6}  }  +  \dfrac{49 + 20 \sqrt{6} }{  2401  - 2400} }

\sf  \:  \:  \:  \implies{{ 49 - 20 \sqrt{6}  }  +49 + 20 \sqrt{6}  }

\sf  \:  \:  \:  \implies{98  }\\

  • \begin{gathered}\\\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\gray{\begin{gathered}\tiny\begin{gathered}\small{\small{\small{\small{\small{\small{\small{\small{\small{\small{\begin{gathered}\begin{gathered}\begin{gathered}\\\\\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\red{ \bigstar} \: \underline{\bf{\orange{More \: Useful \: Formula}}}\\ {\boxed{\begin{array}{cc}\dashrightarrow \sf(a + b)^{2} = {a}^{2} + {b}^{2} + 2ab \\\\\dashrightarrow \sf(a - b)^{2} = {a}^{2} + {b}^{2} - 2ab \\\\\dashrightarrow \sf(a + b)(a - b) = {a}^{2} - {b}^{2} \\\\\dashrightarrow \sf(a + b) ^{3} = {a}^{3} + b^{3} + 3ab(a + b) \\\\ \dashrightarrow\sf(a - b) ^{3} = {a}^{3} - b^{3} - 3ab(a - b) \\ \\\dashrightarrow\sf a ^{3} + {b}^{3} = (a + b)(a ^{2} + {b}^{2} - ab) \\\\\dashrightarrow \sf a ^{3} - {b}^{3} = (a - b)(a ^{2} + {b}^{2} + ab )\\\\\dashrightarrow \sf{a²+b²=(a+b)²-2ab}\\ \end{array}}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}}}}}}}}}}}\end{gathered}\end{gathered}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\\ \end{gathered}
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