Question

2-√5/2+√5=a√5+b find rational number a and b

Answer 1

Hey friend,
Here is the answer you were looking for:
$\frac{2 - \sqrt{5} }{2 + \sqrt{5} } = a \sqrt{5} + b \\ \\ on \: rationalizing \: the \: denominator \: we \: get \\ \\ = \frac{2 - \sqrt{5} }{2 + \sqrt{5} } \times \frac{2 - \sqrt{5} }{2 - \sqrt{5} } \\ \\ using \: the \: identities \\ {(a -b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = \frac{ {(2)}^{2} + {( \sqrt{5}) }^{2} - 2(2)( \sqrt{5} ) }{ {(2)}^{2} - {( \sqrt{5} })^{2} } \\ \\ = \frac{4 + 5 - 4 \sqrt{5} }{4 - 5} \\ \\ = \frac{9 - 4 \sqrt{5} }{ - 1} \\ \\ = - (9 - 4 \sqrt{5} ) \\ \\ = - 9 + 4 \sqrt{5} \\ \\ 4 \sqrt{5} - 9 = a \sqrt{5} + b \\ \\ a = 4 \\ \\ b = - 9$


Hope this helps!!!

@Mahak24

Thanks...
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