`alpha, beta` are roots of `y^(2)-2y-7=0` find,
(i) `alpha^(2)+beta^(2) " (ii) " alpha^(3)+beta^(3)`

Answer

`alpha^(2)+beta^(2)=18` (2) `alpha^(3)+beta^(3)=50`

Explanation

The given equation is `y^(2)-2y-7=0`. Here, a=1, b=-2, c=-7.
`alpha+beta=-( b)/(a)=-(-2)/(1)=2" "......(1)`
`alphabeta=(c)/(a)=-(-7)/(1)=-7" "....(2)`
(1) `alpha^(2)+beta^(2)=(alpha+beta)^(2)-2alphabeta`
`=(2)^(2)-2(-7)" "....` [From (1) and (2)]
`=4+14+18`
(2) `alpha^(3)+beta^(3)=(alpha+beta)^(3)-3alphabeta(alpha+beta)`
`=(2)^(3)-3(-7)(2)" "......` [From (1) and (2)]
=8+42=50