Question

If $a,b,cϵR$ and equations $a{x}^2+bx+c=0$ and $x^2+2x+9=0$ have a common root, show that $a:b:c=1:2:9.$

Answer 1

Given equations are: $x^2+2x+9=0$ (i)
and $a{x}^2+bx+c=0$ (ii)
Clearly, roots of equation (i) are imaginary since equation (i) and (ii) have a common root, therefore common root must be imaginary, also we know that imaginary roots always occur in a pair, hence both roots will be common.
Therefore equations (i) and (ii) are identical
$\therefore$ $\frac{a}{1}=\frac{b}{2}=\frac{c}{9}$
$\Rightarrow$ $a:b:c=1:2:9$