Question

if a^2-5a-1=0, evaluate (1) a^2+1/a^2, (2) a^3-1/a^3​

Answer 1

=> a^2 - 5a - 1 = 0

=> a^2 - 1 = 5a

=> (a^2 - 1)/a = 5a/a {divide by a}

=> a - 1/a = 5 ...(1)

(i) Square on both sides of (1),

=> (a - 1/a)^2 = 5^2

=> a^2 + 1/a^2 - 2 = 25

=> a^2 + 1/a^2 = 27

(ii) Cube on both sides of (1),

=> (a - 1/a)^3 = 5^3

=> a^3 - 1/a^3 - 3(a * 1/a)(a - 1/a) = 125

=> a^3 - 1/a^3 - 3(1)(5) = 125 {a - 1/a = 5}

=> a^3 - 1/a^3 = 125 + 15 = 140