Given
a^2 - 4a - 1 = 0
a^2 - 1 = 4a
a^2 - 1
——— = 4
a
(i . answer ) a - 1/a = 4
( a - 1/a )^2=( 4 )^2 [Squaring both sides]
--> a^2 + 1/a^2 - 2 = 16
--> a^2 + 1/a^2 = 16 + 2
--> a^2 + 1/a^2 = 18 (iv. Answer)
--> a^2 + 1/a^2 + 2 = 18 + 2
--> (a + 1/a)^2 = 20
--> a + 1/a = + or - root 20
--> a + 1/a = + or - 2 root 5 (answer ii.)
a - 1/a = 4 ..............(1)
a + 1/a = + or - 2 root 5............(2)
(1) x (2) =
(a - 1/a)(a + 1/a) = 4 x (+ or - 2 root 5)
a^2 - 1/a^2 = + or - 8 root 5 (answer iii.)
