Question

Find the value of a and b in the following:

`(3sqrt(5) + 1)/(2sqrt(5) + 4) = a + bsqrt(5)`

Answer 1

We are given:

`(3sqrt(5) + 1)/(2sqrt(5) + 4) = a + bsqrt(5)`

Step 1: Multiply numerator and denominator by the conjugate of the denominator:

Conjugate of `2sqrt(5) + 4` is `2sqrt(5) - 4`

`(3sqrt(5) + 1)/(2sqrt(5) + 4) xx (2sqrt(5) - 4)/(2sqrt(5) - 4)`

Step 2: Expand numerator using distributive law:

`(3sqrt(5) + 1)(2sqrt(5) - 4)`

= `3sqrt(5) xx 2sqrt(5) + 3sqrt(5) xx (-4) + 1 xx 2sqrt(5) + 1 xx (-4) `

= `6 xx 5 - 12sqrt(5) + 2sqrt(5) - 4`

= `30 - 10sqrt(5) - 4`

= `26 - 10sqrt(5)`

Step 3: Simplify denominator using (a + b)(a – b) = a² – b²:

`(2sqrt(5))^2 - 4^2`

= 4 × 5 – 16

= 20 – 16

= 4

Step 4: Final expression:

`(26 - 10sqrt(5))/4`

= `26/4 - 10/4 sqrt(5)`

= `13/2 - 5/2 sqrt(5)` 

`a = 13/2, b = -5/2`