To determine which function has zeros at [tex]\( x = 10 \)[/tex] and [tex]\( x = 2 \)[/tex], we need to check each given function step-by-step and see if these values satisfy the conditions for being zero.
### Step-by-Step Solution:
1. Function 1: [tex]\( f(x) = x^2 - 12x + 20 \)[/tex]
- To find out if [tex]\( x = 10 \)[/tex] is a zero:
[tex]\[
f(10) = 10^2 - 12 \cdot 10 + 20 = 100 - 120 + 20 = 0
\][/tex]
So, [tex]\( x = 10 \)[/tex] is a zero of [tex]\( f(x) \)[/tex].
- To find out if [tex]\( x = 2 \)[/tex] is a zero:
[tex]\[
f(2) = 2^2 - 12 \cdot 2 + 20 = 4 - 24 + 20 = 0
\][/tex]
So, [tex]\( x = 2 \)[/tex] is also a zero of [tex]\( f(x) \)[/tex].
Thus, the function [tex]\( f(x) = x^2 - 12x + 20 \)[/tex] has zeros at [tex]\( x = 10 \)[/tex] and [tex]\( x = 2 \)[/tex].
2. Function 2: [tex]\( f(x) = x^2 - 20x + 12 \)[/tex]
- To find out if [tex]\( x = 10 \)[/tex] is a zero:
[tex]\[
f(10) = 10^2 - 20 \cdot 10 + 12 = 100 - 200 + 12 = -88
\][/tex]
Since [tex]\( f(10) \ne 0 \)[/tex], [tex]\( x = 10 \)[/tex] is not a zero of [tex]\( f(x) \)[/tex].
3. Function 3: [tex]\( f(x) = 5x^2 + 40x + 60 \)[/tex]
- To find out if [tex]\( x = 10 \)[/tex] is a zero:
[tex]\[
f(10) = 5 \cdot 10^2 + 40 \cdot 10 + 60 = 500 + 400 + 60 = 960
\][/tex]
Since [tex]\( f(10) \ne 0 \)[/tex], [tex]\( x = 10 \)[/tex] is not a zero of [tex]\( f(x) \)[/tex].
4. Function 4: [tex]\( f(x) = 5x^2 + 60x + 100 \)[/tex]
- To find out if [tex]\( x = 10 \)[/tex] is a zero:
[tex]\[
f(10) = 5 \cdot 10^2 + 60 \cdot 10 + 100 = 500 + 600 + 100 = 1200
\][/tex]
Since [tex]\( f(10) \ne 0 \)[/tex], [tex]\( x = 10 \)[/tex] is not a zero of [tex]\( f(x) \)[/tex].
After checking all the functions, only [tex]\( f(x) = x^2 - 12x + 20 \)[/tex] has zeros at [tex]\( x = 10 \)[/tex] and [tex]\( x = 2 \)[/tex].
Therefore, the function that has zeros at [tex]\( x=10 \)[/tex] and [tex]\( x=2 \)[/tex] is:
[tex]\[ f(x) = x^2 - 12x + 20 \][/tex]
