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Q.
The number of integers $n$ such that the quadratic equation $n x^2+(n+1) x+(n+2)=0$ has rational roots is
Complex Numbers and Quadratic Equations
Solution:
$\Delta=(n+1)^2-4 n(n+2)=-\left(3 n^2+6 n-1\right)=4-3(n+1)^2$
This takes positive values for $n \in\{-2,-1,0\}$.
But $n=0$ is rejected because above equation is given quadratic and in all these two cases, the value is a perfect square.
So, the answer is 2.