The condition for the  polynomial equation $a{x}^2+bx+c=0$  to be a quadratic is

The given equation $a{x}^2+bx+c=0$  is a quadratic equation only if $a\ne 0,$  

for if $a=0$  then the equation becomes $bx+c=0,$  

which is not a second degree equation.

So $a$  must be not equal to zero.