Factorise..a^2 +1/a^2 -3

Answer 1

a^2 +1/a^2 -3
= (a^4 + 1 - 3a^2)/a^2
= (a^4 - 2a^2 + 1 - a^2)/a^2
= ( (a^2 - 1)^2 - a^2)/a^2
= *a^2 - 1 + a)(a^2 - 1 - a)/a^2

Answer 2

or,
a^2 +1/a^2 -3
= a^2 + 2 + 1/a^2 - 5
= (a^2 + 2 + 1/a^2) - 5
= (a + 1/a)^2 - 5
= (a + 1/a + √5)(a + 1/a - √5)

or, continuing with Reiny's solution,
(a^2 - 1 + a)(a^2 - 1 - a)/a^2
= (1 - 1/a^2 + 1/a)(a^2 - 1 - a)
= -(1/a^2 - 1/a - 1)(a^2-a-1)
= -((1/a^2 - 1/a + 1/4) - 5/4)(a^2 - a + 1/4 - 5/4)
= -((1/a - 1/2)^2 - 5/4)((a - 1/2)^2 - 5/4)
= -(√5/2-(1/a-1/2))(√5/2-(a-1/2))(√5/2+(1/a-1/2))(√5/2+(a-1/2))
*whew* !