Question

For what integral value of a are the roots of the equation ax² + (2a − 1)x +(a − 2) = 0 rational?

Answer 1

The roots are rational if the discriminant is a perfect square. That is, (2a − 1)2 − 4(a) (a − 2) must be a perfect square. That is, 4a + 1 must be a perfect square. Since a is an integer, 4a + 1 must be an odd number and a perfect square. 

The square of an odd number is an odd number. Therefore, 4a + 1 = (2n + 1)2 for some n. This gives

a = 4n² + 4n/4 = n(n + 1)

where n is a natural number.