Question

√5-2 / √5+2 - √5+2 / √5-2 = a+b√5​

Answer 1

Given :

$• \bf{ \dfrac{ \sqrt{5} - 2 }{ \sqrt{5} + 2 } - \dfrac{ \sqrt{5} + 2}{ \sqrt{5} - 2 } = a +b \sqrt{5} }$

According to the question,

$\begin{gathered} \leadsto\bf{ \dfrac{ \sqrt{5} - 2 }{ \sqrt{5} + 2 } - \dfrac{ \sqrt{5} + 2}{ \sqrt{5} - 2 } = a +b \sqrt{5} } \\ \\ \leadsto \bf{ \frac{ \sqrt{5} - 2 }{ \sqrt{5} + 2 } \times \frac{ \sqrt{5} - 2}{ \sqrt{5} - 2 } - \frac{ \sqrt{5} + 2}{ \sqrt{5} - 2} \times \frac{ \sqrt{5} + 2}{ \sqrt{5} + 2 } = a + b \sqrt{5} } \\ \\ \leadsto \bf{ \frac{ (\sqrt{5} - 2) {}^{2} }{5 - 4} - \frac{( \sqrt{5} + 2) {}^{2} }{5 - 4} = a + b \sqrt{5} } \\ \\ \leadsto \bf{ \dfrac{5 - 4 \sqrt{5} + 4 - \big(5 + 4 \sqrt{5} + 4 \big)}{1} = a + b \sqrt{5} } \\ \\ \leadsto \bf{ \cancel5 - 4 \sqrt{5} + \cancel4 - \cancel 5 - \cancel 4 \sqrt{5 } - 4 = a + b \sqrt{5} } \\ \\ \leadsto \bf{ - 8 \sqrt{5} = a + b \sqrt{5} } \\ \\ { \underline{ \boxed{ \bf \red{a = 0 \: }}}} \: \: { \underline{ \boxed{ \bf \red{b = - 8 \sqrt{5} }}}}\end{gathered}$

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