The equation xx+12+xx−12=a(a−1) has

xx+12+xx−12=a(a−1) or  xx+1+xx−12−2xx+1xx−1=a(a−1) or  2x2x2−12−2x2x2−1−a(a−1)=0 or  t2−t−a(a−1)=0  where t=2x2x2−1⇒ t=a or 1−a⇒ 2x2x2−1=a or 2x2x2−1=1−a⇒ x=±aa−2 or x=±a−1a+1When a<−1⇒all roots are real12⇒all roots are real