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....?

1 Answer
Apr 4, 2017

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