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 check whether substituting each value for [tex]\( x \)[/tex] makes the function equal to zero.
Given values:
1. [tex]\( x = \frac{1}{3} \)[/tex]
2. [tex]\( x = 3 \frac{1}{3} \)[/tex]
3. [tex]\( x = 6 \frac{1}{3} \)[/tex]
4. [tex]\( x = 9 \frac{1}{3} \)[/tex]
We can rewrite the mixed numbers as improper fractions:
- [tex]\( x = 3 \frac{1}{3} = \frac{10}{3} \)[/tex]
- [tex]\( x = 6 \frac{1}{3} = \frac{19}{3} \)[/tex]
- [tex]\( x = 9 \frac{1}{3} = \frac{28}{3} \)[/tex]
Let's start with evaluating the function at [tex]\( x = \frac{1}{3} \)[/tex]:
1. Check [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. Check [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]
[tex]\[
= 9 \cdot \frac{100}{9} - 54 \cdot \frac{10}{3} - 19
\][/tex]
[tex]\[
= 100 - 180 - 19
\][/tex]
[tex]\[
= -99
\][/tex]
Since [tex]\( f\left( \frac{10}{3} \right) \neq 0 \)[/tex], [tex]\( x = 3 \frac{1}{3} \)[/tex] is not a zero.
3. Check [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]
[tex]\[
= 9 \cdot \frac{361}{9} - 54 \cdot \frac{19}{3} - 19
\][/tex]
[tex]\[
= 361 - 342 - 19
\][/tex]
[tex]\[
= 0
\][/tex]
Since [tex]\( f\left( \frac{19}{3} \right) = 0 \)[/tex], [tex]\( x = 6 \frac{1}{3} \)[/tex] is indeed a zero of the function.
4. Check [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]
[tex]\[
= 9 \cdot \frac{784}{9} - 54 \cdot \frac{28}{3} - 19
\][/tex]
[tex]\[
= 784 - 504 - 19
\][/tex]
[tex]\[
= 261
\][/tex]
Since [tex]\( f\left( \frac{28}{3} \right) \neq 0 \)[/tex], [tex]\( x = 9 \frac{1}{3} \)[/tex] is not a zero.
Conclusion: The value of [tex]\( x \)[/tex] that is a zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 19 \)[/tex] is [tex]\( x = 6 \frac{1}{3} \)[/tex] (which is [tex]\( x = \frac{19}{3} \)[/tex]).
