To determine which value is a zero of the quadratic function [tex]\( f(x) = 4x^2 + 24x + 11 \)[/tex], we need to substitute each given potential zero into the function and see if the result is zero. A zero of a function is a value of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
Let's evaluate the function at each of the given values:
1. Substituting [tex]\( x = -9.25 \)[/tex]:
[tex]\[
f(-9.25) = 4(-9.25)^2 + 24(-9.25) + 11
\][/tex]
[tex]\[
= 4(85.5625) - 222 + 11
\][/tex]
[tex]\[
= 342.25 - 222 + 11
\][/tex]
[tex]\[
= 342.25 - 211
\][/tex]
[tex]\[
= 131.25
\][/tex]
Since [tex]\( f(-9.25) \neq 0 \)[/tex], -9.25 is not a zero of the function.
2. Substituting [tex]\( x = -5.5 \)[/tex]:
[tex]\[
f(-5.5) = 4(-5.5)^2 + 24(-5.5) + 11
\][/tex]
[tex]\[
= 4(30.25) - 132 + 11
\][/tex]
[tex]\[
= 121 - 132 + 11
\][/tex]
[tex]\[
= 121 - 121
\][/tex]
[tex]\[
= 0
\][/tex]
Since [tex]\( f(-5.5) = 0 \)[/tex], -5.5 is a zero of the function.
3. Substituting [tex]\( x = 0.5 \)[/tex]:
[tex]\[
f(0.5) = 4(0.5)^2 + 24(0.5) + 11
\][/tex]
[tex]\[
= 4(0.25) + 12 + 11
\][/tex]
[tex]\[
= 1 + 12 + 11
\][/tex]
[tex]\[
= 24
\][/tex]
Since [tex]\( f(0.5) \neq 0 \)[/tex], 0.5 is not a zero of the function.
4. Substituting [tex]\( x = 3.25 \)[/tex]:
[tex]\[
f(3.25) = 4(3.25)^2 + 24(3.25) + 11
\][/tex]
[tex]\[
= 4(10.5625) + 78 + 11
\][/tex]
[tex]\[
= 42.25 + 78 + 11
\][/tex]
[tex]\[
= 131.25
\][/tex]
Since [tex]\( f(3.25) \neq 0 \)[/tex], 3.25 is not a zero of the function.
So, the zero of the quadratic function [tex]\( f(x) = 4x^2 + 24x + 11 \)[/tex] from the given options is:
[tex]\[ x = -5.5 \][/tex]
