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.

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