Factorise..a^2 +1/a^2 -3
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
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* !
To factorize the expression a^2 + 1/a^2 - 3, we can follow these steps:
Step 1: Identify the common denominator.
The common denominator of a^2 and 1/a^2 is a^2. So, we can rewrite the expression as (a^4 + 1 - 3a^2) / a^2.
Step 2: Expand the numerator.
The numerator can be expanded to a^4 + 1 - 3a^2.
Step 3: Rearrange the terms.
Rearrange the terms in descending powers of 'a': a^4 - 3a^2 + 1.
Step 4: Try to factorize the expression.
We can treat this expression as a quadratic equation in terms of a^2. The quadratic equation can be factored as (a^2 - 1)(a^2 - 1).
Step 5: Simplify the expression.
(a^2 - 1)(a^2 - 1) can be further simplified as (a - 1)(a + 1)(a - 1)(a + 1).
So, the factored form of the expression a^2 + 1/a^2 - 3 is (a - 1)(a + 1)(a - 1)(a + 1).