α and β are the roots of the quadratic equation x2 – x – 1 = 0. What is the value of α2 + β2?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. 47
α and β are the roots of the quadratic equation x2 – x – 1 = 0
α + β = -(-1)/1 = 1 ....(1)
αβ = -1/1 = -1 .....(2)
β = -1/α
From eq(1)
α + β = 1
α - 1/α = 1
α2 - 1/α2 = (1)2 + 2 = 3
α4 + 1/α4 = (3)2 - 2 = 7
α8 + 1/α8 = (7)2 - 2 = 47
Hence, α8 + 1/α8 = 47