Question

If $x^2+3x+5=0$ and $a{x}^2+bx+c=0$ have a common root and $a,b,c\in N$ then minimum value of $a+b+c$ is equal to:

• A. $9$
• B. $6$
• C. $12$
• D. $3$

Answer 1

The correct option is A

$9$

$x^2+3x+5=0$ has imaginary roots which occur complex conjugate pairs.

$\therefore x^2+3x+5=0,a{x}^2+bx+c$ have both roots in common.

$\therefore \frac{a}{1}=\frac{b}{3}=\frac{c}{5}=\lambda$

$\Rightarrow a+b+c=9\lambda ,a,b,c\in N\Rightarrow \lambda \in N$

$\Rightarrow min.(a+b+c)=9$