To find the zeroes of the function [tex]\( y = -2(x - 6k)(x + 2k) \)[/tex], we need to determine the values of [tex]\( x \)[/tex] that make [tex]\( y \)[/tex] equal to zero. When [tex]\( y = 0 \)[/tex], the equation becomes:
[tex]\[ 0 = -2(x - 6k)(x + 2k) \][/tex]
We can set each factor inside the parentheses to zero and solve for [tex]\( x \)[/tex] separately.
1. Set [tex]\( x - 6k = 0 \)[/tex]:
[tex]\[ x - 6k = 0 \][/tex]
[tex]\[ x = 6k \][/tex]
2. Set [tex]\( x + 2k = 0 \)[/tex]:
[tex]\[ x + 2k = 0 \][/tex]
[tex]\[ x = -2k \][/tex]
Therefore, the zeroes of the function are [tex]\( x = 6k \)[/tex] and [tex]\( x = -2k \)[/tex].
Comparing this with the given choices:
(1) [tex]\( x = -6k \)[/tex] and [tex]\( x = 2k \)[/tex]
(2) [tex]\( x = -12k \)[/tex] and [tex]\( x = 4k \)[/tex]
(3) [tex]\( x = -k \)[/tex] and [tex]\( x = 3k \)[/tex]
(4) [tex]\( x = -2k \)[/tex] and [tex]\( x = 6k \)[/tex]
The correct choice that matches our zeroes [tex]\( x = 6k \)[/tex] and [tex]\( x = -2k \)[/tex] is:
(4) [tex]\( x = -2k \)[/tex] and [tex]\( x = 6k \)[/tex]
So, the correct answer is choice (4).
