Question

If a² - 5a - 1 = 0 and a ≠ 0 ; find:

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

Answer 1

(i) Consider the given equation

a² - 5a - 1 = 0

Rewrite the given equation, we have

a² - 1 = 5a

⇒ `[ a^2 - 1 ]/a = 5`

⇒ `[ a^2/a - 1/a ] = 5`

⇒ `a - 1/a = 5`                       ...(1)

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

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

∴ `( a - 1/a )^2 = a^2 + 1/a^2 - 2`

⇒ `(5)^2 = a^2 + 1/a^2 - 2`                  [From(1)]

⇒ `25 = a^2 + 1/a^2 - 2`

⇒ `a^2 + 1/a^2 = 27`                ...(2)

Now consider the identity (a + b)² = a² + b² + 2ab

∴ `( a + 1/a )^2 = a^2 + 1/a^2 + 2`

⇒ `( a + 1/a )^2 = 27 + 2`              [From (2)]

⇒ `( a + 1/a )^2 = 29`     

⇒ `a + 1/a = +- sqrt29`              ...(3)

(iii) We need to find `a^2 - 1/a^2`

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

∴ `a^2 - 1/a^2 = ( a + 1/a )( a - 1/a )`              ...(4)

From equation (3), we have,

` a + 1/a = +- sqrt29`

From equation (1), we have,

`a - 1/a = 5`;

Thus, identity (4), becomes,

`a^2 - 1/a^2 = (+- sqrt29)(5)`

⇒ `a^2 - 1/a^2 = 5(+- sqrt29 )`