Question 164
There are real numbers $\alpha$, $\beta$, such that the cubic functions
$$
f(x) = x^3 - 3x^2 + \alpha x + \beta \quad \text{and}
$$
$$
g(x) = x^3 + (\alpha - 3)x - 6
$$
have exactly two distinct non-zero roots in common. Find $\alpha$ and $\beta$.
$$
f(x) = x^3 - 3x^2 + \alpha x + \beta \quad \text{and}
$$
$$
g(x) = x^3 + (\alpha - 3)x - 6
$$
have exactly two distinct non-zero roots in common. Find $\alpha$ and $\beta$.