To determine which of the given values is a zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 19 \)[/tex], we need to test each value by substituting it into the function and checking if the result is zero.
We will test the values [tex]\( x = \frac{1}{3} \)[/tex], [tex]\( x = 3 \frac{1}{3} \)[/tex], [tex]\( x = 6 \frac{1}{3} \)[/tex], and [tex]\( x = 9 \frac{1}{3} \)[/tex].
1. Testing [tex]\( x = \frac{1}{3} \)[/tex]:
[tex]\[
f\left(\frac{1}{3}\right) = 9\left(\frac{1}{3}\right)^2 - 54\left(\frac{1}{3}\right) - 19
\][/tex]
[tex]\[
= 9 \cdot \frac{1}{9} - 54 \cdot \frac{1}{3} - 19
\][/tex]
[tex]\[
= 1 - 18 - 19
\][/tex]
[tex]\[
= -36
\][/tex]
Since [tex]\( f\left(\frac{1}{3}\right) \neq 0 \)[/tex], [tex]\( x = \frac{1}{3} \)[/tex] is not a zero.
2. Testing [tex]\( x = 3 \frac{1}{3} = \frac{10}{3} \)[/tex]:
[tex]\[
f\left(3 \frac{1}{3}\right) = 9\left(\frac{10}{3}\right)^2 - 54\left(\frac{10}{3}\right) - 19
\][/tex]
[tex]\[
= 9 \cdot \frac{100}{9} - 54 \cdot \frac{10}{3} - 19
\][/tex]
[tex]\[
= 100 - 180 - 19
\][/tex]
[tex]\[
= -99
\][/tex]
Since [tex]\( f\left(3 \frac{1}{3}\right) \neq 0 \)[/tex], [tex]\( x = 3 \frac{1}{3} \)[/tex] is not a zero.
3. Testing [tex]\( x = 6 \frac{1}{3} = \frac{19}{3} \)[/tex]:
[tex]\[
f\left(6 \frac{1}{3}\right) = 9\left(\frac{19}{3}\right)^2 - 54\left(\frac{19}{3}\right) - 19
\][/tex]
[tex]\[
= 9 \cdot \frac{361}{9} - 54 \cdot \frac{19}{3} - 19
\][/tex]
[tex]\[
= 361 - 342 - 19
\][/tex]
[tex]\[
= 0
\][/tex]
Since [tex]\( f\left(6 \frac{1}{3}\right) = 0 \)[/tex], [tex]\( x = 6 \frac{1}{3} \)[/tex] is indeed a zero.
4. Testing [tex]\( x = 9 \frac{1}{3} = \frac{28}{3} \)[/tex]:
[tex]\[
f\left(9 \frac{1}{3}\right) = 9\left(\frac{28}{3}\right)^2 - 54\left(\frac{28}{3}\right) - 19
\][/tex]
[tex]\[
= 9 \cdot \frac{784}{9} - 54 \cdot \frac{28}{3} - 19
\][/tex]
[tex]\[
= 784 - 504 - 19
\][/tex]
[tex]\[
= 261
\][/tex]
Since [tex]\( f\left(9 \frac{1}{3}\right) \neq 0 \)[/tex], [tex]\( x = 9 \frac{1}{3} \)[/tex] is not a zero.
Therefore, the correct zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 19 \)[/tex] is [tex]\( x = 6 \frac{1}{3} \)[/tex].
