Integers between roots!

Algebra Level 3

Find the sum of all possible values of $a \in Z$ such that exactly two integers lie between the roots of the equation $x^2 + ax-1$ .

1 solution

Ankit Kumar Jain
Feb 17, 2018

The condition that is necessary and but not sufficient for this to happen is :

Difference of roots lies between 1 and 3 (inclusive)

$\Rightarrow 1 < \sqrt{a^2 + 4} \leq 3 \Rightarrow 1 < a^2+4 \leq 9 \Rightarrow a^2 \leq 5$

Possible values of $a \in Z$ are $-2 , -1 , 0 , 1, 2$

But verifying the solutions gives only $-1 , 1$ as the solutions.

ANSWER = $\boxed{0}$