Question

If alpha,beta are the roots of the equation x²-2x+4=0 then find the value of alpha^n+beta^n?

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Answer 1

Given info : α and β are the roots of the equation x² - 2x + 4 = 0.

To find : the value of αⁿ + βⁿ

solution : here, x² - 2x + 4 = 0

Discriminant, D = b² - 4ac = (-2)² - 4(4) = 4 - 16 = -12 < 0

so roots are imaginary.

using formula of finding roots,

x = {-b ± √D}/2a

= {-(-2) ±√(-12)}/2

= (2 ± 2√3i)/2 [ as we know, √-1 = i ]

= 1 ± √3i

let α = 1 + √3i and β = 1 - √3i

now αⁿ + βⁿ

= (1 + √3i)ⁿ + (1 - √3i)ⁿ

= (2)ⁿ[cosπ/3 + isinπ/3]ⁿ + (2)ⁿ[cosπ/3 - isinπ/3]ⁿ

= 2ⁿ [cosnπ/3 + isinnπ/3] + 2ⁿ[cosnπ/3 - isinnπ/3]

= 2ⁿ[cosnπ/3 + isinnπ/3 + cosnπ/3 - isinnπ/3]

= 2ⁿ[2cosnπ/3]

=2ⁿ⁺¹ cosnπ/3

Therefore the value of αⁿ + βⁿ = 2ⁿ⁺¹ cos(nπ/3)

Answer 2

Answer:

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