Question 162
There are real numbers $\alpha, \beta, \gamma$ such that the cubic functions
$$
f(x) = x^3 - 3x^2 + \alpha x + \beta \quad \text{and} \quad g(x) = x^3 + \gamma x - 6
$$
have exactly two distinct non-zero roots in common. Prove that those two roots satisfy the equation $3(\alpha - \gamma - 9)x^2 + (\alpha - \gamma - 9)(\gamma - \alpha)x + 9\beta = 0$. Also prove that necessary conditions on $\alpha, \beta, \gamma$ are $(\alpha - \gamma)(\beta + 6) = 54$ and $(\alpha - \gamma)^3 + 9\gamma(\alpha - \gamma) + 162 = 0$.
$$
f(x) = x^3 - 3x^2 + \alpha x + \beta \quad \text{and} \quad g(x) = x^3 + \gamma x - 6
$$
have exactly two distinct non-zero roots in common. Prove that those two roots satisfy the equation $3(\alpha - \gamma - 9)x^2 + (\alpha - \gamma - 9)(\gamma - \alpha)x + 9\beta = 0$. Also prove that necessary conditions on $\alpha, \beta, \gamma$ are $(\alpha - \gamma)(\beta + 6) = 54$ and $(\alpha - \gamma)^3 + 9\gamma(\alpha - \gamma) + 162 = 0$.