Question

If a² - 3a + 1 = 0, and a ≠ 0; find: 

1. `a + 1/a`         
2. `a^2 + 1/a^2`

Answer 1

(i) Consider the given equation

a² - 3a + 1 = 0

Rewrite the given equation, we have

a² + 1 = 3a

⇒ `[ a^2 + 1 ]/a = 3`

⇒ `[ a^2/a + 1/a ] = 3`

⇒ `[ a + 1/a ] = 3`           ...(1)

(ii) We need to find `a^2 + 1/a^2`:

We know the identity, (a + b)² = a² + b² + 2ab

∴ `(a + 1/a )^2 = a^2 + 1/a^2 + 2`           ...(2)

From equation (1), we have,

`a + 1/a` = 3

Thus, equation (2), becomes,

⇒ `(3)^2 = a^2 + 1/a^2 + 2`

⇒ 9 = `a^2 + 1/a^2 + 2`

⇒ `a^2 + 1/a^2 = 7`