If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2 + bx + c = 0$, construct a quadratic equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.

Find the condition that the quadratic equations $a_1 x^2 + b_1 x + c_1 = 0$ and $a_2 x^2 + b_2 x + c_2 = 0$ have a common root.

Determine the value(s) of $k$ for which the quadratic equation $$(k + 1)x^2 + 2(k + 3)x + (k + 8) = 0$$ has equal roots. Find the roots for each value of $k$.