Only one of the roots of $a{x}^2+bx+c=0,a\ne 0$  is zero if

The given equation is $a{x}^2+bx+c=0$ ……. ( 1 )

When one root is 0 and the other is non zero,

then sum of the roots is non-zero and product is zero

from ( 1 ):

$\Rightarrow -\frac{b}{a}\ne 0and\frac{c}{a}=0$

$\Rightarrow b\ne 0$  and $c=0.$