3+√5/2√5+3=a+b√5

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Solution

(3+root5)/ (2root5+3) = a+b root5

LHS =(3+root5) / (2root5+3)

= (3+root5)×(2root5-3) / (2root5+3)×(2root5-3)

=6root5-9+10-3root5 / (2root5)^2 - (3)^2

=1+3root5 / 20-9

=1+3root5/11

=1/11 + 3root5/11

LHS = RHS
1/11 + 3root5/11 = a+b root5

hence,
a = 1/11 and b = 3/11

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