Question

If the equations $x^2+3x+5=0$ and $a{x}^2+bx+c=0;a,b,c\in N$ have a common root, then the least possible value of $a+b+c$ is

• A. $3$
• B. $1$
• C. $5$
• D. $9$

Answer 1

The correct option is D $9$

$x^2+3x+5=0$
$\Delta =9-20=-11<0$
Thus, the above equation has imaginary roots.
The coefficients are real, so the imaginary roots will be in conjugate pair.

Given $x^2+3x+5=0$ and $a{x}^2+bx+c=0$ have a common root.
So, we can conclude that both the roots are common.
$\Rightarrow \frac{a}{1}=\frac{b}{3}=\frac{c}{5}=k\left(\text{say}\right)$
$\Rightarrow a+b+c=9k$

But given that $a,b,c\in N$
$\therefore$ The minimum value of $a+b+c$ is $9$