If α is a complex constant such that αz2+z+α=0 has a real root, then

Let z = c be a real root. Then,αc2+c+α=0                     (1)Putting α=p+iq, we have(p+iq)c2+c+p−iq=0⇒pc2+c+p=0 and qc2−q=0⇒c=±1                         (∵q≠0)∴(1)⇒α±1+α=0Also, |c| = 1