If ax2+(b−c)x+a−b−c=0 has unequal real roots for all c∈R, then
b < 0 < a
a < 0 < b
b < a < 0
b > a > 0
We have,
D=(b−c)2−4a(a−b−c)>0 or b2+c2−2bc−4a2+4ab+4ac>0 or c2+(4a−2b)c−4a2+4ab+b2>0 for all c∈R
Discriminant of the above expression in c must be negative.
Hence,
(4a−2b)2−4−4a2+4ab+b2<0 or 4a2−4ab+b2+4a2−4ab−b2<0 or a(a−b)<0⇒ a<0 and a−b>0 or a>0 and a−b<0⇒ b<a<0 or b>a>0