Question

If $x^2+3x+5=0$ and $a{x}^2+bx+c=0$ have a common root, $a,b,cϵN$, find the minimum value of a+b+c.

Answer 1

Let $p,q$ be the two roots. Now, by Vieta's formulas,

$p+q=-ba-3$ and

$pq=ca=5$

$\therefore b=3a,c=5a$.

Hence, $a+b+c=a+3a+5a=9a$.

But given that a,b,c∈N, therefore, the minimum value of $a+b+c=9$.