F4 A.maths Root-Coefficient3

If the roots of 3x2 + 3x – 1 = 0 are α ,β,evaluate the following expressions.
Sum of roots α + β = -3/3 = -1
Product of roots αβ = -1/3
1. 1/α + 1/β
= (α + β)/ αβ
= -1 / (-1/3)
= 3
2. β/α + α/β
= (β2 + α2) / αβ
= (α2 + 2αβ + β2 – 2αβ) / αβ
= [(α + β)2 – 2αβ] / αβ
= (1 + 2/3) / (-1/3)
= -5
3. 1/(α + 1) + 1/(β + 1)
= [(β + 1) + (α + 1)]/[(α + 1)(β + 1)]
= (α + β + 2)/(αβ + α + β + 1)
= (-1 + 2) / (-1/3 – 1 + 1)
= 1 / (-1/3)
= -3