Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
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]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
