Question

If `(5sqrt2−e)(3sqrt2+e)=fsqrt2-6`, where `e` and `f` are positive integers, find `e` and `f`?

Answer 1

`e=6` and `f=12`

Explanation:

As `(5sqrt2−e)(3sqrt2+e)=fsqrt2-6`

i.e. `5sqrt2(3sqrt2+e)-e(3sqrt2+e)=fsqrt2-6`

or `30+5esqrt2-3esqrt2-e^2=fsqrt2-6`

or `30-e^2+2esqrt2=fsqrt2-6`

Hence `30-e^2=-6` i.e. `e^2=36` and as `e` is a positive integer

we have `e=6`

and `2esqrt2=fsqrt2`

i.e. `f=2e=12`