Question

If a+1/a=3,find the values of (a-1/a)^2.

Answer 1

Answer:

(a + 1/a)² = 5

Note:

• (a+b)² = a² + 2ab + b²

• (a-b)² = a² - 2ab + b²

• (a + b)(a - b) = a² - b²

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

Solution:

• Given : a + 1/a = 3
• To Find : (a - 1/a)² = ?

We have ;

a + 1/a = 3

Now,

Squaring both sides , we get ;

=> (a + 1/a)² = 3²

=> a² + 2•a•(1/a) + (1/a)² = 9

=> a² + 2 + 1/a² = 9

=> a² + 1/a² = 9 - 2

=> a² + 1/a² = 7

Now,

=> (a - 1/a)² = a² - 2•a•(1/a) + (1/a)²

=> (a - 1/a)² = a² - 2 + 1/a²

=> (a - 1/a)² = (a² + 1/a²) - 2

=> (a - 1/a)² = 7 - 2 { a² + 1/a² = 7 }

=> (a - 1/a)² = 5

Hence ,

The required answer is 5 .

Using direct formula :

(a - b)² = (a + b)² - 4ab

=> (a - 1/a)² = (a + 1/a)² - 4•a•(1/a)

=> (a - 1/a)² = 3² - 4

=> (a - 1/a)² = 9 - 4

=> (a - 1/a)² = 5

Thus,

(a - 1/a)² = 5