To determine which value is a zero of the quadratic function [tex]\( f(x) = 16x^2 + 32x - 9 \)[/tex], we need to evaluate the function at each of the given possible zeros and see if it equals zero.
Let’s evaluate the function step-by-step for each given value:
1. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = -5.25 \)[/tex]
[tex]\[
f(-5.25) = 16(-5.25)^2 + 32(-5.25) - 9
\][/tex]
Calculate [tex]\( (-5.25)^2 \)[/tex]:
[tex]\[
(-5.25)^2 = 27.5625
\][/tex]
Substitute back into the function:
[tex]\[
f(-5.25) = 16(27.5625) + 32(-5.25) - 9
\][/tex]
Calculate each term:
[tex]\[
16 \cdot 27.5625 = 441
\][/tex]
[tex]\[
32 \cdot (-5.25) = -168
\][/tex]
Add the terms together:
[tex]\[
f(-5.25) = 441 - 168 - 9 = 264
\][/tex]
Since [tex]\( f(-5.25) \neq 0 \)[/tex], [tex]\( x = -5.25 \)[/tex] is not a zero of the function.
2. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = -2.25 \)[/tex]
[tex]\[
f(-2.25) = 16(-2.25)^2 + 32(-2.25) - 9
\][/tex]
Calculate [tex]\( (-2.25)^2 \)[/tex]:
[tex]\[
(-2.25)^2 = 5.0625
\][/tex]
Substitute back into the function:
[tex]\[
f(-2.25) = 16(5.0625) + 32(-2.25) - 9
\][/tex]
Calculate each term:
[tex]\[
16 \cdot 5.0625 = 81
\][/tex]
[tex]\[
32 \cdot (-2.25) = -72
\][/tex]
Add the terms together:
[tex]\[
f(-2.25) = 81 - 72 - 9 = 0
\][/tex]
Since [tex]\( f(-2.25) = 0 \)[/tex], [tex]\( x = -2.25 \)[/tex] is indeed a zero of the function.
3. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = -125 \)[/tex]
[tex]\[
f(-125) = 16(-125)^2 + 32(-125) - 9
\][/tex]
Calculate [tex]\( (-125)^2 \)[/tex]:
[tex]\[
(-125)^2 = 15625
\][/tex]
Substitute back into the function:
[tex]\[
f(-125) = 16(15625) + 32(-125) - 9
\][/tex]
Calculate each term:
[tex]\[
16 \cdot 15625 = 250000
\][/tex]
[tex]\[
32 \cdot (-125) = -4000
\][/tex]
Add the terms together:
[tex]\[
f(-125) = 250000 - 4000 - 9 = 245991
\][/tex]
Since [tex]\( f(-125) \neq 0 \)[/tex], [tex]\( x = -125 \)[/tex] is not a zero of the function.
4. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = -0.25 \)[/tex]
[tex]\[
f(-0.25) = 16(-0.25)^2 + 32(-0.25) - 9
\][/tex]
Calculate [tex]\( (-0.25)^2 \)[/tex]:
[tex]\[
(-0.25)^2 = 0.0625
\][/tex]
Substitute back into the function:
[tex]\[
f(-0.25) = 16(0.0625) + 32(-0.25) - 9
\][/tex]
Calculate each term:
[tex]\[
16 \cdot 0.0625 = 1
\][/tex]
[tex]\[
32 \cdot (-0.25) = -8
\][/tex]
Add the terms together:
[tex]\[
f(-0.25) = 1 - 8 - 9 = -16
\][/tex]
Since [tex]\( f(-0.25) \neq 0 \)[/tex], [tex]\( x = -0.25 \)[/tex] is not a zero of the function.
From all the evaluations, the value that makes the function equal to zero is [tex]\( x = -2.25 \)[/tex]. Thus, the correct answer is:
[tex]\( x = -2.25 \)[/tex]
