Answer :
To determine which, if any, of the given values are zeros of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 19 \)[/tex], we need to substitute each value into the function and check if the result equals zero.
### Step-by-step verification:
1. For [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]\[ = 9 \cdot \frac{1}{9} - 18 - 19 \][/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 of the function.
2. For [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]\[ = 9 \cdot \frac{100}{9} - 180 - 19 \][/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 of the function.
3. For [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]\[ = 9 \cdot \frac{361}{9} - 342 - 19 \][/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 a zero of the function.
4. For [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]\[ = 9 \cdot \frac{784}{9} - 504 - 19 \][/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 of the function.
### Conclusion:
The zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 19 \)[/tex] among the given options is:
[tex]\[ x = 6 \frac{1}{3} \][/tex]
### Step-by-step verification:
1. For [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]\[ = 9 \cdot \frac{1}{9} - 18 - 19 \][/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 of the function.
2. For [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]\[ = 9 \cdot \frac{100}{9} - 180 - 19 \][/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 of the function.
3. For [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]\[ = 9 \cdot \frac{361}{9} - 342 - 19 \][/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 a zero of the function.
4. For [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]\[ = 9 \cdot \frac{784}{9} - 504 - 19 \][/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 of the function.
### Conclusion:
The zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 19 \)[/tex] among the given options is:
[tex]\[ x = 6 \frac{1}{3} \][/tex]
