Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine which values are zeroes of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 192 \)[/tex], we will evaluate the function at each of the given points and check if the result is zero. Let's go through each candidate one by one:
1. For [tex]\( x = \frac{1}{3} \)[/tex]:
[tex]\[ f\left(\frac{1}{3}\right) = 9\left(\frac{1}{3}\right)^2 - 54\left(\frac{1}{3}\right) - 192 \][/tex]
Simplify inside the parentheses:
[tex]\[ f\left(\frac{1}{3}\right) = 9 \left(\frac{1}{3} \times \frac{1}{3}\right) - 54 \times \frac{1}{3} - 192 \][/tex]
[tex]\[ = 9 \times \frac{1}{9} - 18 - 192 \][/tex]
[tex]\[ = 1 - 18 - 192 \][/tex]
[tex]\[ = -209 \][/tex]
Thus, [tex]\( f\left(\frac{1}{3}\right) \neq 0 \)[/tex].
2. For [tex]\( x = 3 \frac{1}{3} = \frac{10}{3} \)[/tex]:
[tex]\[ f\left(\frac{10}{3}\right) = 9\left(\frac{10}{3}\right)^2 - 54\left(\frac{10}{3}\right) - 192 \][/tex]
Simplify inside the parentheses:
[tex]\[ f\left(\frac{10}{3}\right) = 9 \left(\frac{100}{9}\right) - 54 \times \frac{10}{3} - 192 \][/tex]
[tex]\[ = 9 \times \frac{100}{9} - 180 - 192 \][/tex]
[tex]\[ = 100 - 180 - 192 \][/tex]
[tex]\[ = -272 \][/tex]
Thus, [tex]\( f\left(\frac{10}{3}\right) \neq 0 \)[/tex].
3. For [tex]\( x = 6 \frac{1}{3} = \frac{19}{3} \)[/tex]:
[tex]\[ f\left(\frac{19}{3}\right) = 9\left(\frac{19}{3}\right)^2 - 54\left(\frac{19}{3}\right) - 192 \][/tex]
Simplify inside the parentheses:
[tex]\[ f\left(\frac{19}{3}\right) = 9 \left(\frac{361}{9}\right) - 54 \times \frac{19}{3} - 192 \][/tex]
[tex]\[ = 9 \times \frac{361}{9} - 342 - 192 \][/tex]
[tex]\[ = 361 - 342 - 192 \][/tex]
[tex]\[ = -173 \][/tex]
Thus, [tex]\( f\left(\frac{19}{3}\right) \neq 0 \)[/tex].
4. For [tex]\( x = 9 \frac{1}{3} = \frac{28}{3} \)[/tex]:
[tex]\[ f\left(\frac{28}{3}\right) = 9\left(\frac{28}{3}\right)^2 - 54\left(\frac{28}{3}\right) - 192 \][/tex]
Simplify inside the parentheses:
[tex]\[ f\left(\frac{28}{3}\right) = 9 \left(\frac{784}{9}\right) - 54 \times \frac{28}{3} - 192 \][/tex]
[tex]\[ = 9 \times \frac{784}{9} - 504 - 192 \][/tex]
[tex]\[ = 784 - 504 - 192 \][/tex]
[tex]\[ = 88 \][/tex]
Thus, [tex]\( f\left(\frac{28}{3}\right) \neq 0 \)[/tex].
Since none of the given values satisfy [tex]\( f(x) = 0 \)[/tex], we conclude that none of them is a zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 192 \)[/tex]. Therefore, the answer is:
```
None
```
1. For [tex]\( x = \frac{1}{3} \)[/tex]:
[tex]\[ f\left(\frac{1}{3}\right) = 9\left(\frac{1}{3}\right)^2 - 54\left(\frac{1}{3}\right) - 192 \][/tex]
Simplify inside the parentheses:
[tex]\[ f\left(\frac{1}{3}\right) = 9 \left(\frac{1}{3} \times \frac{1}{3}\right) - 54 \times \frac{1}{3} - 192 \][/tex]
[tex]\[ = 9 \times \frac{1}{9} - 18 - 192 \][/tex]
[tex]\[ = 1 - 18 - 192 \][/tex]
[tex]\[ = -209 \][/tex]
Thus, [tex]\( f\left(\frac{1}{3}\right) \neq 0 \)[/tex].
2. For [tex]\( x = 3 \frac{1}{3} = \frac{10}{3} \)[/tex]:
[tex]\[ f\left(\frac{10}{3}\right) = 9\left(\frac{10}{3}\right)^2 - 54\left(\frac{10}{3}\right) - 192 \][/tex]
Simplify inside the parentheses:
[tex]\[ f\left(\frac{10}{3}\right) = 9 \left(\frac{100}{9}\right) - 54 \times \frac{10}{3} - 192 \][/tex]
[tex]\[ = 9 \times \frac{100}{9} - 180 - 192 \][/tex]
[tex]\[ = 100 - 180 - 192 \][/tex]
[tex]\[ = -272 \][/tex]
Thus, [tex]\( f\left(\frac{10}{3}\right) \neq 0 \)[/tex].
3. For [tex]\( x = 6 \frac{1}{3} = \frac{19}{3} \)[/tex]:
[tex]\[ f\left(\frac{19}{3}\right) = 9\left(\frac{19}{3}\right)^2 - 54\left(\frac{19}{3}\right) - 192 \][/tex]
Simplify inside the parentheses:
[tex]\[ f\left(\frac{19}{3}\right) = 9 \left(\frac{361}{9}\right) - 54 \times \frac{19}{3} - 192 \][/tex]
[tex]\[ = 9 \times \frac{361}{9} - 342 - 192 \][/tex]
[tex]\[ = 361 - 342 - 192 \][/tex]
[tex]\[ = -173 \][/tex]
Thus, [tex]\( f\left(\frac{19}{3}\right) \neq 0 \)[/tex].
4. For [tex]\( x = 9 \frac{1}{3} = \frac{28}{3} \)[/tex]:
[tex]\[ f\left(\frac{28}{3}\right) = 9\left(\frac{28}{3}\right)^2 - 54\left(\frac{28}{3}\right) - 192 \][/tex]
Simplify inside the parentheses:
[tex]\[ f\left(\frac{28}{3}\right) = 9 \left(\frac{784}{9}\right) - 54 \times \frac{28}{3} - 192 \][/tex]
[tex]\[ = 9 \times \frac{784}{9} - 504 - 192 \][/tex]
[tex]\[ = 784 - 504 - 192 \][/tex]
[tex]\[ = 88 \][/tex]
Thus, [tex]\( f\left(\frac{28}{3}\right) \neq 0 \)[/tex].
Since none of the given values satisfy [tex]\( f(x) = 0 \)[/tex], we conclude that none of them is a zero of the quadratic function [tex]\( f(x) = 9x^2 - 54x - 192 \)[/tex]. Therefore, the answer is:
```
None
```
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.