Question

If p=3-2root2, find the value of p²+1/p²​

Answer 1

Given :-

• p = 3 - 2√2

To find :-

• p² + 1/p²

Solution :-

Firstly we have to find 1/p

• p = 3 - 2√2

→ 1/p = 1/3 - 2√2

Rationalize its denominator

→ 1/p = 1/3 - 2√2 × 3 + 2√2/3 + 2√2

Apply identity a² - b² = (a + b)(a - b)

→ 1/p = 3 + 2√2/(3)² - (2√2)²

→ 1/p = 3 + 2√2/9 - 8

→ 1/p = 3 + 2√2

According to the given condition

→ p² + 1/p²

→ (3 - 2√2)² + (3 + 2√2)²

Apply identity :

• (a - b)² = a² + b² - 2ab
• (a + b)² = a² + b² + 2ab

→ [(3)² + (2√2)² - 2 × 3 × 2√2] + [(3)² + (2√2)² + 2 × 3 × 2√2]

→ [9 + 8 - 12√2] + [9 + 8 + 12√2]

→ 9 + 8 - 12√2 + 9 + 8 + 12√2

→ 9 + 8 + 9 + 8

→ 17 + 17

→ 34

Hence,

• p² + 1/p² = 34