Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our 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
```
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
