if alpha and beta are roots of x square-2x+5=0, find the equation
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Answer:
α,β are the roots of x
2
−2x+4=0
⇒α+β=2,αβ=4
α
2
+β
2
=(α+β)
2
−2αβ
=2
2
−2(4)
=4−8=−4
α
3
+β
3
=(α+β)
3
−3αβ(α+β)
=2
3
−3(4)(2)
=8−24=−16
Now, α
5
+β
5
=(α
3
+β
3
)(α
2
+β
2
)−α
3
β
2
−α
2
β
3
=(α
3
+β
3
)(α
2
+β
2
)−(αβ)
2
(α+β)
=(−16)(−4)−(4)
2
(2)
=64−32=32
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