If a^2-2a-1=0. Then find a-1/a,a+1a,a^2-1/a^2?

a² - 2a - 1 = 0
(a² - 2a - 1)/a = 0 ...... for a ≠ 0
a - 2 - (1/a) = 0
a - (1/a) = 2

[a - (1/a)]² = 2²
a² - 2 + (1/a)² = 4
a² - 2 + (1/a)² + 4 = 4 + 4
a² - 2 + (1/a)² + 4 = 4 + 4
a² + 2 + (1/a)² = 8
a² + 2a(1/a) + (1/a)² = 8
[a + (1/a)]² = 8
√[a + (1/a)]² = √8
a + (1/a) = 2√2 or a + (1/) = -2√2

[a - (1/a)] [a + (1/a)] = 2 × 2√2 or [a - (1/a)] [a + (1/a)] = 2 × (-2√2)
a² - (1/a²) = 4√2 or a² - (1/a²) = -4√2

Parentheses please

a = 1 + sqrt(2) , 1 - sqrt(2)
a-1/a = 2
a+1/a = ((1+sqrt(2))^2 + 1)/(1+sqrt(2))
Similarly calculate a^2 - 1/a^2