Let $k$ be a real number and let $f(x) := x^{2} + 2(k-2)x + k$. Find the range of $k$ if $\alpha > \beta$ are such that $f(x) = 0$ for $x = \alpha, \beta$ and if $\beta < -1 < \alpha.$
The given answer is $k < 1.$ However, I suspect it is wrong. For, we have $$\beta = -k+2 - \sqrt{(k-1)(k-4)} < -1 < -k+2 + \sqrt{(k-1)(k-4)} = \alpha,$$ and both inequalities lead to $k > 5.$
