If $2b^2 + 27d = 9bc$, find the relationship using roots.
Step-by-Step Solution
Key Concept: Express symmetric functions of roots in terms of coefficients and use the given constraint to solve for the required expression.
Let $\alpha, \beta, \gamma$ be the required roots. From the given condition $\alpha + \beta + \gamma = -b$, $\alpha\beta + \beta\gamma + \gamma\alpha = c$, and $\alpha\beta\gamma = -d$. Also, $\alpha + \gamma = 2\beta$ gives $2\beta = 3\beta/3 = -b$. From $\alpha + \gamma = 2\beta$ and $\alpha\beta\gamma + \gamma\alpha = c$, we obtain $\beta(\alpha + \gamma) + \gamma\alpha = c$, leading to $\beta^2 + \alpha\gamma = c$. Substituting into the constraint: $\frac{9b}{c} + \frac{34}{c} = c$, giving $2b^3 + 27d = 9bc$, hence the answer is $90$.
Correct Answer: 90