given that α and β are the roots of the equation x²=x+7 (i) Prove that (a) 1 / α = α-1 / 7 (b)α³=8α+7 (ii)Find the numerical value of α/β + β/α
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Step-by-step explanation:
ANSWER
x
2
+2x+2=0
Here, a=1,b=2,,c=2
From quadratic formula,
x=
2×1
−2±
2
2
−4×2×1
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⇒x=
2
−2±2i
=−1±i
Therefore,
α=−1+i⇒α
2
=(−1+i)
2
=−2i
β=−1−i⇒β
2
=(−1−i)
2
=2i
Now,
α
15
+β
15
=(α
2
)
7
⋅α+(β
2
)
7
⋅β
=(−2i)
7
(−1+i)+(2i)
7
(−1−i)
=(−2)
7
(−i)(−1+i)+2
7
(−i)(−1−i)
=2
7
i(−1+i)+2
7
i(1+i)
=2
7
i(−1+i+1+i)
=2
7
i⋅2i
=2
8
⋅(−1)
=−256
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