Question

If a = 1 - √2 , find the value of (a −$\frac{1}{a})^{2}$

Answer 1

Solution :-

Here , given that

a = 1 - √2

We need to find ,

[a - (1/a)]²

Simplifying the given question using the identity

• (a - b)² = a² + b² - 2ab

Similarly

→ (a - 1/a)² = a² + (1/a)² - 2(a)(1/a)

→ (a - 1/a)² = a² + 1/a² - 2

Now , substituting the value of a

→ (1 - √2)² + 1/(1 - √2)² - 2

It can be written as

→ (1 - √2)² + 1/(1 - √2)² - 2(1 - √2)[1/(1 - √2)]

It is in the form of

• a² + b² - 2ab = (a - b)²

Now , similarly

→ [ (1 - √2) - 1/(1 - √2) ]²

→ (1 - √2)² - 1/(1 - √2)

→ 1 - (√2)² - 2(1)(√2) - 1/(1 - √2)

→ 2 - 2√2/(1 - √2)

Taking common

→ 2(1 - √2)/(1 - √2)

→ 2

Hence , (a - 1/a)² = 2.