Let's find which value of [tex]\( x \)[/tex] among the given options makes the quadratic function [tex]\( f(x) = 4x^2 + 24x + 11 \)[/tex] equal to zero. That is, we need to find the root of the equation:
[tex]\[ 4x^2 + 24x + 11 = 0.\][/tex]
We will evaluate the function at each given value to see which one satisfies the equation [tex]\( f(x) = 0 \)[/tex].
1. Evaluate at [tex]\( x = -9.25 \)[/tex]:
[tex]\[
f(-9.25) = 4(-9.25)^2 + 24(-9.25) + 11
= 4(85.5625) + 24(-9.25) + 11
= 342.25 - 222 + 11
= 131.25
\][/tex]
[tex]\( f(-9.25) \neq 0 \)[/tex].
2. Evaluate at [tex]\( x = -5.5 \)[/tex]:
[tex]\[
f(-5.5) = 4(-5.5)^2 + 24(-5.5) + 11
= 4(30.25) + 24(-5.5) + 11
= 121 - 132 + 11
= 0
\][/tex]
[tex]\( f(-5.5) = 0 \)[/tex].
3. Evaluate at [tex]\( x = 0.5 \)[/tex]:
[tex]\[
f(0.5) = 4(0.5)^2 + 24(0.5) + 11
= 4(0.25) + 24(0.5) + 11
= 1 + 12 + 11
= 24
\][/tex]
[tex]\( f(0.5) \neq 0 \)[/tex].
4. Evaluate at [tex]\( x = 3.25 \)[/tex]:
[tex]\[
f(3.25) = 4(3.25)^2 + 24(3.25) + 11
= 4(10.5625) + 24(3.25) + 11
= 42.25 + 78 + 11
= 131.25
\][/tex]
[tex]\( f(3.25) \neq 0 \)[/tex].
After evaluating the function at all given values, we find that [tex]\( x = -5.5 \)[/tex] is the only value that makes [tex]\( f(x) = 0 \)[/tex].
Thus, the zero of the quadratic function [tex]\( f(x) = 4x^2 + 24x + 11 \)[/tex] is:
[tex]\[ x = -5.5. \][/tex]
