Question

Pe multimea numerelor reale se defineste legea de compozitie asociativa x*y=4(x+y-3)-xy
1.aratati ca x*y=4-(x-4)(y-4),pt. orice nr.reale x,y
2.rezolvati in multimea nr.reale ecuatia x*x*x=x

Answer 1

1.   x×y = 4x+4y-12 -xy = ( 4x -xy ) + (4y-16) +4
=   4 + [ -x( y- 4 ) + 4( y-4 ) ]= 4 + [ ( y-4 )( -x+ 4 ) ]
= 4  - ( x-4) ( y-4)

2. x×x= 4- ( x- 4) ²
[x×x]×x = [ 4 - ( x-4) ² ] ×x=  4 - [4 - (x-4)²- 4](x -4) =4 + ( x-4 )³
4+ ( x -4)³= x
(x-4)³ - ( x-4 ) =0
( x-4) [(x-4)² -1]=0
(x-4) ( x-4-1) ( x-4 +1) =0
x=4    x=5       x=3