Question

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

Answer 1

Given

• x = 5 - 2√6

To Find

1. x+1/x
2. x² + 1/x²

Solution

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

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

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

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

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

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

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

___________________

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

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

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

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

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

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

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

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

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

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

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

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