Finde eine geeignete Substitution und löse: (4 - (5 - x ) )^3 = (8 - 3 (5 - x))^2

Question

(4 - (5 - x ) )³ = (8 - 3 (5 - x))²

Answer 1

(4 - (5 - x))^3 = (8 - 3·(5 - x))^2

z = 5 - x

(4 - z)^3 = (8 - 3·z)^2

- z^3 + 12·z^2 - 48·z + 64 = 9·z^2 - 48·z + 64

- z^3 + 3·z^2 = 0

z^2·(- z + 3) = 0

z = 0 ; x = 5

z = 3 ; x = 2

Answer 2

( 4 - ( 5 - x ) ) ³ = ( 8 - 3 ( 5 - x ) ) ²

Setze K := 5 - x

Dann:

( 4 - K ) ³ = ( 8 - 3 K ) ²

<=> 64 - 48 K + 12 K ² - K ³ = 64 - 48 K + 9 K ²

<=>  K³ - 3 K ²  = 0

<=> K ² ( K - 3 ) = 0

<=> K = 0 ( doppelte Nullstelle) oder K = 3

Rücksubstitution:

<=> 5 - x = 0 oder 5 - x = 3

<=> x = 5 (doppelte Nullstelle) oder x = 2