Question

Let a cubic equation $x ^3+ ax ^2+ bx + c =0$ has roots $\alpha, \beta, \gamma$. If 2 is added to each, then the results are reciprocal of square of original roots, then

• A. $a + b + c =1 $
• B. $a +2 b + c =1$
• C. $ abc = a +2 b +2 c$
• D. $abc =2 a +2 b + c$

Answer 1

Answer: (A), (B), (C)

$\because \alpha+2=\frac{1}{\alpha^2} \Rightarrow \alpha^3+2 \alpha^2-1=0$
$\because x ^3+ ax ^2+ bx + c =0$ and $x ^3+2 x ^2-1=0$ have all roots in common.
$\therefore a =2, b =0, c =-1$.