F4 根與系數 roots and coefficient

x^2 + (k + 2)x + k = 0
Sum of roots α + β = - k - 2 ... (A)
Product of roots αβ = k ... (B)
1/α + 1/β
= (α + β)/ αβ
Use (A) and (B)
= (-k - 2)/k = -1/3
3k + 6 = k
k = -3
The equation is x^2 - x - 3 = 0
Since α and β are roots of the equation
α^2 - α - 3 = 0 ... (1)
β^2 - β - 3 = 0 ... (2)
(1) => α^2 = α + 3
α^3 = α^2 + 3α
α^3 = (α + 3) + 3α
α^3 = 4α + 3 ... (3)
(2) => β^2 = β + 3
4β^2 = 4β + 12 ... (4)
(3) + (4) => α^3 + 4β^2 = 4α + 3 + 4β + 12
α^3 + 4β^2 = 4(α + β) + 15
α^3 + 4β^2 = 4 + 15 = 19