Question

the expression y= ax^2+bx +c has always the same sign as a if

Answer 1

there are two cases
for a<0 the discriminant need to be less than 0

for a>0 also the discriminant needs to be less than 0

Answer 2

y = ax² + bx + c always being same sign
this is possible only when, y = ax² + bx + c doesn't cut any point in the x-axis .
so, two case will be possible ,

case 1 :- when, graph is vertically upward . means a > 0 and D < 0

case 2 : - when, graph is vertically downward . means a < 0 and D < 0

see the graph