Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Our platform offers a seamless experience for finding reliable answers from a network of experienced professionals. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our 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]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
