Question

If the roots of the equation $x^{3}+bx^{2}+cx+d=0$ are in arithmetic progression, then $b, \, c$ and $d$ satisfy the relation

• A. $2b^{2}-27d=9bc$
• B. $2b^{3}-27d=9bc$
• C. $2b^{2}+27d=9bc$
• D. $2b^{3}+27d=9bc$

Answer 1

Answer: (D)

Let $\alpha ,\beta ,\gamma $ be the required roots
$\alpha +\beta +\gamma =-b, \, \alpha \beta +\beta \gamma +\gamma \alpha =c$ , $\alpha \beta \gamma =-d$
Also, $\alpha +\gamma =2\beta \Rightarrow 3\beta =-b$
$\beta \left(\right.\alpha +\gamma \left.\right)+\gamma \alpha =c\Rightarrow \left(\frac{- b}{3}\right)2\left(\frac{- b}{3}\right)+\gamma \alpha =c$
$\Rightarrow \frac{2 b^{2}}{9}-\frac{d}{\beta }=c\Rightarrow \frac{2 b^{2}}{9}-\frac{d}{- \frac{b}{3}}=c$
$\Rightarrow \frac{2 b^{2}}{9}+\frac{3 d}{b}=c\Rightarrow 2b^{3}+27d=9bc$