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Algebra

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Evaluate a^3 - \dfrac{1}{a^3} if a^2 - a - 1 = 0.

Jun 5, 2024

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a^3   - 1/a^3

a^2 - a -1 = 0

a^2 -a + 1/4 = 1  + 1/4

(a - 1/2)^2  = 5/4            take both roots

a -1/2  = sqrt 5/2         or   a -1  = -sqrt 5/2

a = [1 + sqrt 5]/2          a = [ 1 -sqrt 5]/2

a^3  = [ 1 +sqrt 5] /2  *  [1 + sqrt 5]/2  *[1 +sqrt 5] /2

a^3  = [ 1 + 2sqrt 5 + 5 ] / 4   *  [ 1 + sqrt 5 ] /2

a^3 = [ 1 + 2sqrt 5 + 5 + sqrt 5 + 10 + 5sqrt 5] / 8

a^3  = [16 + 8sqrt 5] / 8  =  2 + sqrt 5

1/a^3  =   1/ [2 + sqrt 5]  =  [2-sqrt 5] / [4-5]  = sqrt 5 -2

So in one case  a^3 - 1/a^3  =  [2 + sqrt 5] -[ sqrt 5  - 2]  =  4

In the other case

a^3 = [ 1 -sqrt 5]/2 * [1 -sqrt 5]/2 * [1 -sqrt 5]/2

a^3  = [ 1 - 2sqrt 5 + 5]/4 * [ 1 -sqrt 5]/2

a^3  =  [ 1 -2sqrt 5 + 5 - sqrt 5 +10 -5sqrt 5] /8

a^3  = [ 16 - 8sqrt 5]/8 = 2 -sqrt 5

1/a^3  =  1/[2 - sqrt 5]  =  [2 + sqrt 5] / [4-5] = -2-sqrt 5

So in the other case  a^3 - 1/a^3 =  [2-sqrt 5 ] -  [- 2 - sqrt 5 ] =  4     (same result !!!)

Jun 5, 2024
edited by CPhill  Jun 5, 2024
edited by CPhill  Jun 5, 2024