a) x2 - 4x + y2 - 6y + 13
= ( x2 - 4x + 4 ) + ( y2 - 6y + 9 )
= ( x - 2 )2 + ( y - 3 )2
b) x2 - 2xy + 2y2 + 2y + 1
= ( x2 - 2xy + y2 ) + ( y2 + 2y + 1 )
= ( x - y )2 + ( y + 1 )2
c) 4x2 - 12x - y2 + 2y + 8
= ( 4x2 - 12x + 9 ) - ( y2 - 2y + 1 )
= ( 2x - 3 )2 - ( y - 1 )2
= [ ( 2x - 3 ) - ( y - 1 ) ][ ( 2x - 3 ) + ( y - 1 ) ]
= ( 2x - 3 - y + 1 )( 2x - 3 + y - 1 )
= ( 2x - y - 2 )( 2x + y - 4 )
d) x2 + y2 + z2 - 6x - 4y - 2z + 14
= ( x2 - 6x + 9 ) + ( y2 - 4y + 4 ) + ( z2 - 2z + 1 )
= ( x - 3 )2 + ( y - 2 )2 + ( z - 1 )2
