Quadratic Equations

The quadratic equation $x^2 - (k-1)x + (k+2) = 0$ has roots $\alpha$ and $\beta$. If $\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{1}{4}$, find the possible values of $k$.

Determine the value(s) of the constant $k$ such that the quadratic equation $kx^2 - 4x + 1 = 0$ has exactly one real solution.

Given that $k > 0$ and the quadratic equation $x^2 - 2kx + (k^2 - 1) = 0$ satisfies that the product of its roots is 4 times the difference of its roots, find the real roots of the equation.