To determine which of the given options is a zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 19 \)[/tex], we need to check each option one by one and see if it satisfies the equation [tex]\( f(x) = 0 \)[/tex].
We will evaluate [tex]\( f(x) \)[/tex] at each proposed solution and check if the result is zero.
1. Option: [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]
Simplifying further:
[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. Option: [tex]\( x = 3\frac{1}{3} = \frac{10}{3} \)[/tex]
[tex]\[
f\left(\frac{10}{3}\right) = 9\left(\frac{10}{3}\right)^2 - 54\left(\frac{10}{3}\right) - 19
\][/tex]
Simplifying further:
[tex]\[
= 9 \cdot \left(\frac{100}{9}\right) - 54 \cdot \frac{10}{3} - 19
\][/tex]
[tex]\[
= 100 - 180 - 19
\][/tex]
[tex]\[
= -99
\][/tex]
Since [tex]\( f\left(\frac{10}{3}\right) \neq 0 \)[/tex], [tex]\( x = 3\frac{1}{3} \)[/tex] is not a zero.
3. Option: [tex]\( x = 6\frac{1}{3} = \frac{19}{3} \)[/tex]
[tex]\[
f\left(\frac{19}{3}\right) = 9\left(\frac{19}{3}\right)^2 - 54\left(\frac{19}{3}\right) - 19
\][/tex]
Simplifying further:
[tex]\[
= 9 \cdot \left(\frac{361}{9}\right) - 54 \cdot \frac{19}{3} - 19
\][/tex]
[tex]\[
= 361 - 342 - 19
\][/tex]
[tex]\[
= 0
\][/tex]
Here, [tex]\( f\left(\frac{19}{3}\right) = 0 \)[/tex], so [tex]\( x = 6\frac{1}{3} \)[/tex] is a zero of the quadratic function.
4. Option: [tex]\( x = 9\frac{1}{3} = \frac{28}{3} \)[/tex]
[tex]\[
f\left(\frac{28}{3}\right) = 9\left(\frac{28}{3}\right)^2 - 54\left(\frac{28}{3}\right) - 19
\][/tex]
Simplifying further:
[tex]\[
= 9 \cdot \left(\frac{784}{9}\right) - 54 \cdot \frac{28}{3} - 19
\][/tex]
[tex]\[
= 784 - 504 - 19
\][/tex]
[tex]\[
= 261
\][/tex]
Since [tex]\( f\left(\frac{28}{3}\right) \neq 0 \)[/tex], [tex]\( x = 9\frac{1}{3} \)[/tex] is not a zero.
Thus, among the given options, the zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 19 \)[/tex] is:
[tex]\[
x = 6\frac{1}{3}
\][/tex]
