Question

If equations $x^2+bx+c=0$ and $b{x}^2+cx+1=0$ have a common root then

• A. $b+c+1=0$
• B. $b^2+c^2+1=bc$
• C. $(b-c{)}^2+(c-1{)}^2+(b-1{)}^2=0$
• D. $b+c+1=-bc$

Answer 1

Answer: (A), (C)

The correct options are
A $b+c+1=0$
C $(b-c{)}^2+(c-1{)}^2+(b-1{)}^2=0$

Let $\alpha$ be the common root.
${\alpha }^2+b\alpha +c=0$
$b{\alpha }^2+c\alpha +1=0$
So $(b^2-c)\alpha +bc-1=0$
$\alpha =\frac{1-bc}{b^2-c}$
So putting $\alpha$ in equation
$(1-bc{)}^2+b(1-bc\left)\right(b^2-c)+c(b^2-c{)}^2=0$
$\Rightarrow b^3+c^3+1-3bc=0$
$\Rightarrow (b+c+1)\left(\right(b-c{)}^2+(c-1{)}^2+(b-1{)}^2)=0$
$\Rightarrow (b+c+1)=0\text{or}(b-c{)}^2+(c-1{)}^2+(b-1{)}^2=0$