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Question
Find the value of a and b in the following:
`(3 - sqrt(5))/(3 + 2sqrt(5)) = asqrt(5) - 19/11`
Solution
We have, `(3 - sqrt(5))/(3 + 2sqrt(5)) = asqrt(5) - 19/11`
For rationalising the above equation, we multiply numerator and denominator of LHS by `3 - 2sqrt(5)`, we get
⇒ `((3 - sqrt(5)))/(3 + 2sqrt(5)) xx (3 - 2sqrt(5))/(3 - 2sqrt(5)) = asqrt(5) - 19/11`
⇒ `(3(3 - 2sqrt(5)) - sqrt(5)(3 - 2sqrt(5)))/((3)^2 - (2sqrt(5))^2) = asqrt(5) - 19/11` ...[Using identity, (a – b)(a + b) = a2 – b2]
⇒ `(9 - 6sqrt(5) - 3sqrt(5) + 10)/(9 - 4 xx 5) = asqrt(5) - 19/11`
⇒ `(19 - 9sqrt(5))/(9 - 20) = asqrt(5) - 19/11`
⇒ `(19 - 9sqrt(5))/(-11) = asqrt(5) - 19/11`
⇒ `(9sqrt(5))/11 - 19/11 = asqrt(5) - 19/11`
⇒ `(9sqrt(5))/11 = asqrt(5)`
⇒ `a = 9/11`
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