Let's analyze which quadratic function [tex]\( h(x) \)[/tex] has the given roots [tex]\( x = 3 \)[/tex] and [tex]\( x = -10 \)[/tex]. Since [tex]\( x = 3 \)[/tex] and [tex]\( x = -10 \)[/tex] are roots of the quadratic function, we can write [tex]\( h(x) \)[/tex] in its factored form:
[tex]\[
h(x) = k(x - 3)(x + 10)
\][/tex]
where [tex]\( k \)[/tex] is a constant coefficient.
We need to determine which of the given functions matches this structure. Here are the candidates:
1. [tex]\( h(x) = x^2 - 13x - 30 \)[/tex]
2. [tex]\( h(x) = x^2 - 7x - 30 \)[/tex]
3. [tex]\( h(x) = 2x^2 + 26x - 60 \)[/tex]
4. [tex]\( h(x) = 2x^2 + 14x - 60 \)[/tex]
### Step-by-Step Solution:
Candidate 1: [tex]\( h(x) = x^2 - 13x - 30 \)[/tex]
We can evaluate [tex]\( h(x) \)[/tex] at [tex]\( x = 3 \)[/tex] and [tex]\( x = -10 \)[/tex]:
[tex]\[
h(3) = 3^2 - 13(3) - 30 = 9 - 39 - 30 = -60 \neq 0
\][/tex]
Thus, [tex]\( h(x) = x^2 - 13x - 30 \)[/tex] does not satisfy [tex]\( h(3) = 0 \)[/tex] and [tex]\( h(-10) = 0 \)[/tex].
Candidate 2: [tex]\( h(x) = x^2 - 7x - 30 \)[/tex]
We can evaluate [tex]\( h(x) \)[/tex] at [tex]\( x = 3 \)[/tex] and [tex]\( x = -10 \)[/tex]:
[tex]\[
h(3) = 3^2 - 7(3) - 30 = 9 - 21 - 30 = -42 \neq 0
\][/tex]
Thus, [tex]\( h(x) = x^2 - 7x - 30 \)[/tex] does not satisfy [tex]\( h(3) = 0 \)[/tex] and [tex]\( h(-10) = 0 \)[/tex].
Candidate 3: [tex]\( h(x) = 2x^2 + 26x - 60 \)[/tex]
We can evaluate [tex]\( h(x) \)[/tex] at [tex]\( x = 3 \)[/tex] and [tex]\( x = -10 \)[/tex]:
[tex]\[
h(3) = 2(3^2) + 26(3) - 60 = 2(9) + 78 - 60 = 18 + 78 - 60 = 36 \neq 0
\][/tex]
Thus, [tex]\( h(x) = 2x^2 + 26x - 60 \)[/tex] does not satisfy [tex]\( h(3) = 0 \)[/tex] and [tex]\( h(-10) = 0 \)[/tex].
Candidate 4: [tex]\( h(x) = 2x^2 + 14x - 60 \)[/tex]
We can evaluate [tex]\( h(x) \)[/tex] at [tex]\( x = 3 \)[/tex] and [tex]\( x = -10 \)[/tex]:
[tex]\[
h(3) = 2(3^2) + 14(3) - 60 = 2(9) + 42 - 60 = 18 + 42 - 60 = 0
\][/tex]
[tex]\[
h(-10) = 2(-10^2) + 14(-10) - 60 = 2(100) + (-140) - 60 = 200 - 140 - 60 = 0
\][/tex]
Since [tex]\( h(3) = 0 \)[/tex] and [tex]\( h(-10) = 0 \)[/tex], [tex]\( h(x) = 2x^2 + 14x - 60 \)[/tex] satisfies the given conditions.
Therefore, the function that represents [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = 2x^2 + 14x - 60
\][/tex]
