To determine which of the given quadratic functions has zeros at [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex], let's analyze each function step by step.
### Quadratic Function Zeros
A quadratic function [tex]\( f(x) \)[/tex] has the form:
[tex]\[ f(x) = a(x - r_1)(x - r_2) \][/tex]
where [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex] are the zeros (roots) of the function. Given zeros are [tex]\( r_1 = 6 \)[/tex] and [tex]\( r_2 = -4 \)[/tex]. Thus, the function can be written as:
[tex]\[ f(x) = a(x - 6)(x + 4) \][/tex]
where [tex]\( a \)[/tex] is a non-zero constant.
### Analysis of Each Choice
Let’s check each function one by one to see which has the zeros at 6 and -4.
#### Choice A:
[tex]\[ f(x) = (x - 6)(x + 4) \][/tex]
- At [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = (6 - 6)(6 + 4) = 0 \cdot 10 = 0 \][/tex]
- At [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = (-4 - 6)(-4 + 4) = (-10) \cdot 0 = 0 \][/tex]
Since [tex]\( f(x) = 0 \)[/tex] at both [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex], function A is a candidate.
#### Choice B:
[tex]\[ f(x) = (x + 6)(x - 4) \][/tex]
- At [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = (6 + 6)(6 - 4) = 12 \cdot 2 = 24 \neq 0 \][/tex]
Since [tex]\( f(x) \neq 0 \)[/tex] at [tex]\( x = 6 \)[/tex], function B is not a candidate.
#### Choice C:
[tex]\[ f(x) = (x - 6)(x - 4) \][/tex]
- At [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = (6 - 6)(6 - 4) = 0 \cdot 2 = 0 \][/tex]
- At [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = (-4 - 6)(-4 - 4) = (-10)(-8) = 80 \neq 0 \][/tex]
Since [tex]\( f(x) \neq 0 \)[/tex] at [tex]\( x = -4 \)[/tex], function C is not a candidate.
#### Choice D:
[tex]\[ f(x) = (x + 6)(x + 4) \][/tex]
- At [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = (6 + 6)(6 + 4) = 12 \cdot 10 = 120 \neq 0 \][/tex]
Since [tex]\( f(x) \neq 0 \)[/tex] at [tex]\( x = 6 \)[/tex], function D is not a candidate.
### Conclusion
The function [tex]\( f(x) = (x - 6)(x + 4) \)[/tex] is the only one that has zeros at [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
Thus, the correct choice is:
[tex]\[ \boxed{A} \][/tex]
