x-2-4x-1-0-are-roots-3-17-5-solution-is-root-2-4-1-0-2-4-1-3-17-5-2-17-5-

Question Number 162174 by mnjuly1970 last updated on 27/Dec/21

x^{2} - 4 x - 1 = 0

α, β a r e r o o t s

α^{3} + 17 β + 5 = ?

- - - s o l u t i o n - - -

α i s r o o t \Rightarrow α^{2} - 4 α - 1 = 0

\Rightarrow α^{2} = 4 α + 1 ✓

α^{3} + 17 β + 5 = α . α^{2} + 17 β + 5

= α (4 α + 1) + 17 β + 5

= 4 α^{2} + α + 17 β + 5

= 4 (4 α + 1) + α + 17 β + 5 = 17 (α + β) + 9

= 17 S + 9 = 17 (4) + 9 = 77

Commented by cortano last updated on 27/Dec/21

x^{2} - 4 x - 1 = 0 {\begin{cases} α \\ β \end{cases} \Rightarrow α + β = 4

\Rightarrow α^{2} = 4 α + 1

\Rightarrow α^{3} = 4 α^{2} + α

\Rightarrow α^{3} + 17 β + 5 = 4 α^{2} + α + 17 β + 5

= 4 (4 α + 1) + 16 β + 9

= 16 (α + β) + 13

= 64 + 13 = 77

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