Question

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

Answer 1

$x^2 +3x+5=0$
\(\Delta =9-20=-11 \lt 0\)
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 $ax^2 +bx+c=0$ have a common root.
So, we can conclude that both the roots are common.
$\Rightarrow \dfrac a1=\dfrac b3=\dfrac c5=k~(\text{say})$
$\Rightarrow a+b+c = 9k$

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