Question

Integers a and b are such that `(a + 3sqrt(5))^2` + `a - b sqrt(5)` = 51. Find the possible values of a and corresponding values of b....?

Answer 1

`a = -3` and `b=-18` or `a=2` and `b=12`

Explanation:

`(a+3sqrt5)^2+a-bsqrt5 = 51` if and only if

`a^2+6asqrt5 + 45 +a-bsqrt5 = 51`. Which is equivalent to

`a^2+a+45 +(6a -b)sqrt5 = 51`

Notice that the right hand side is rational, in fact it is an integer. It does not include any `sqrt5`

On the left, since we are told that `a` and `b` are integers, we must have

`a^2+a+45` is an integer and `(6a -b)sqrt5` is irrational unless `6a-b = 0`.

If the sum is to be equal to `51`, then we must have

`a^2+a+45 = 51` `" "` and `" "` `6a -b = 0`

So `a = -3` or `2` `" "` and `" "` `b=6a`