41
α and β are the roots of the quadratic equation x2 - x - 1 = 0. What is the value of α8 + β8?
- A47
- B54
- C59
- D68
Answer not set
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0
CCommunity
Product of roots, α × β = -1
Sum of roots, α + β = -b/a
⇒ α + β = 1 (? b = -1 and a = 1)
Squaring on both the sides,
⇒ (α + β)2 = 12
⇒ α2 + β2 + 2αβ = 1
⇒ α2 + β2 + 2(-1) = 1
⇒ α2 + β2 = 3
Again squaring on both the sides,
⇒ (α2 + β2)2 = 32
⇒ α4 + β4 + 2α2β2 = 9
⇒ α4 + β4 = 7
Again squaring on both the sides,
⇒ (α4 + β4)2 = 72
⇒ α8 + β8 + 2α4β4 = 49
∴ α8 + β8 = 47
Sum of roots, α + β = -b/a
⇒ α + β = 1 (? b = -1 and a = 1)
Squaring on both the sides,
⇒ (α + β)2 = 12
⇒ α2 + β2 + 2αβ = 1
⇒ α2 + β2 + 2(-1) = 1
⇒ α2 + β2 = 3
Again squaring on both the sides,
⇒ (α2 + β2)2 = 32
⇒ α4 + β4 + 2α2β2 = 9
⇒ α4 + β4 = 7
Again squaring on both the sides,
⇒ (α4 + β4)2 = 72
⇒ α8 + β8 + 2α4β4 = 49
∴ α8 + β8 = 47
0
CCommunity
Product of roots, α × β = -1
Sum of roots, α + β = -b/a
⇒ α + β = 1 (? b = -1 and a = 1)
Squaring on both the sides,
⇒ (α + β)2 = 12
⇒ α2 + β2 + 2αβ = 1
⇒ α2 + β2 + 2(-1) = 1
⇒ α2 + β2 = 3
Again squaring on both the sides,
⇒ (α2 + β2)2 = 32
⇒ α4 + β4 + 2α2β2 = 9
⇒ α4 + β4 = 7
Again squaring on both the sides,
⇒ (α4 + β4)2 = 72
⇒ α8 + β8 + 2α4β4 = 49
∴ α8 + β8 = 47
Sum of roots, α + β = -b/a
⇒ α + β = 1 (? b = -1 and a = 1)
Squaring on both the sides,
⇒ (α + β)2 = 12
⇒ α2 + β2 + 2αβ = 1
⇒ α2 + β2 + 2(-1) = 1
⇒ α2 + β2 = 3
Again squaring on both the sides,
⇒ (α2 + β2)2 = 32
⇒ α4 + β4 + 2α2β2 = 9
⇒ α4 + β4 = 7
Again squaring on both the sides,
⇒ (α4 + β4)2 = 72
⇒ α8 + β8 + 2α4β4 = 49
∴ α8 + β8 = 47