The determinant abaα+bbcbα+caα+bbα+c0=0 is equal to zero, if

Given that abaα+bbcbα+caα+bbα+c0=0Operating C3→C3−C1α−C2, we getab0bc0aα+bbα+c−aα2+bα+bα+c=0⇒ aα2+2bα+cab0bc0aα+bbα+c1=0⇒ ac−b2aα2+2bα+c=0So, either ac−b2=0 or aα2+2bα+c=0This means that either a, b, c are in G.P. or (x−α) is a factor of ax2+2bx+c.