If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$ If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$
Using the condition for common root, $$(3c-5b)(b-3a)=(c-5a)^2$$
$$3bc-9ac-5b^2+15ab=c^2+25a^2-10ac$$
$$25a^2+5b^2+c^2=15ab+3bc+ac$$
There is pattern in the equation. The coefficients of the middle terms and first terms of both sides are $(5\times 5,5)$ and $(5\times 3,3)$.
I tried to use Lagrange multipliers. But isnt there any simpler way to minimize $a+b+c$?

The polynomial $x^2+3x+5$ is irreducible over $\mathbb{Q}$.  Hence, if $x^2+3x+5$ and $ax^2+bx+c$, with $a,b,c\in\mathbb{Q}$, have a common root, they must be proportional.  That is, $$ax^2+bx+c=a\left(x^2+3x+5\right)\,.$$
The problem would be more challenging if $x^2+3x+5$ is replaced by $x^2+3x+2$.
