Question

Please find the answer , it is a very importatnt question

Answer 1

Explanation:

The given expression is :

`(2-\sqrt5)/(2+3\sqrt5)=\sqrt5 x+y` ....(1)

Taking LHS of the above expression

`(2-\sqrt5)/(2+3\sqrt5)`

Rationalizing both denominator and numerator.

`(2-\sqrt5)/(2+3\sqrt5)* (2-3\sqrt5)/(2-3\sqrt5)=(4-6\sqrt5-2\sqrt5+15)/(2^2-(3\sqrt5)^2)\\\\=(19-8\sqrt5)/(4-45)\\\\=(19-8\sqrt5)/(-41)\\\\=(\sqrt5 * 8)/(41)+((-19)/(41))`

Equation (1) becomes,

`(\sqrt5 * 8)/(41)+((-19)/(41))=\sqrt 5\ x+y`

Equating both sides,

`x=(8)/(41)` and
`y=(-19)/(41)`

Hence, this is the required solution.