Question

If $2x^2+5x+7=0$ and $a{x}^2+bx+c=0$ have at least one root common such that $a,b,c\in \{1,2,3,\dots ,100\}$, then the difference between the maximum and the minimum possible value of $a+b+c$ is

Answer 1

Let $f\left(x\right)=2x^2+5x+7$
and $g\left(x\right)=a{x}^2+bx+c$
For $f\left(x\right)=2x^2+5x+7=0$, $D=-31<0$
So, the roots of $f\left(x\right)=0$ are non-real and are in conjugate pair.
Since, the coefficients are real, $f\left(x\right)=2x^2+5x+7=0$ and $g\left(x\right)=a{x}^2+bx+c=0$ will have both roots common.
$\therefore \frac{a}{2}=\frac{b}{5}=\frac{c}{7}=k$
$\Rightarrow a=2k,b=5k,c=7k$

Lowest possible value of $a+b+c$ will occur when $k=1$
When $k=1,$
$a+b+c=2+5+7=14$
Highest possible value of $a+b+c$ will occur when $k=14$
$(\because 7k\le 100\Rightarrow k\le \frac{100}{7})$
When $k=14,$
$a+b+c=28+70+98=196$
Difference between the maximum and the minimum is $196-14=182$