The expression y=ax2+bx+c has always the same sign as c if

Let f(x)=ax2+bx+c. Then f(0)=c. Thus the graph of y=f(x) meets y-axis at (0,c).If c>0, then by hypothesis f(x)>0 This means that the curve y=f(x) does not meet x-axis.If c<0, then by hypothesis f(x)<0, which means that the curve y=f(x) is always below x-axis and so it does not intersect with x-axis. Thus in both cases y=f(x) does not intersect with x-axis i.e. f(x)≠0 for any real x.Hence f(x)=0 i.e., ax2+bx+c=0 has imaginary roots and so b2<4ac