Question

If p=5-2root 2 then find p square +1/p square

Answer 1

given, P = 5 - 2√2

so, 1/P = 1/(5 - 2√2)

= 1/(5 - 2√2) × (5 + 2√2)/(5 + 2√2)

= (5 + 2√2)/{(5 - 2√2)(5 + 2√2)}

= (5 + 2√2)/{5² - (2√2)²}

= (5 + 2√2)/(25 - 8)

= (5 + 2√2)/17

now, P² + 1/P² = (P + 1/P)² - 2 × P × 1/P

= (P + 1/P)² - 2

= {(5 - 2√2) + (5 + 2√2)/17}² - 2

= {(17(5 - 2√2) + 5 + 2√2}²/17² - 2

= {90 - 32√2}²/289 - 2

= {(90 - 32√2)² - 2 × 289}/289

={8100+2048 - 5760√2 - 578}/289

= {9570-5760√2}/289

hence, answer is (9570 - 5760√2)/289