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Question

If α and β are the roots of the equation x22x+4=0, α9+β9=

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Solution

x22x+4=0α,β are roots.
α+β=2
αβ=4
Now, α9+β9=(α3+β3)33α3β3(α3+β3)
=[(α+β)33αβ(α+β)]33(αβ)3((α+β)33αβ(α+β))
=[233(4)(2)]33(4)3((2)33(4)(2))
=(824)3192(824)
=4096+3072
=1024
=29

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