Question
Use a graphing utility to find or to approximate the x-intercepts of the graph of the function.
yequals5xsquaredminus14xplus8
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your answer.
A.
The x-intercept(s) is/are xequals
enter your response here.
(Simplify your answer. Use a comma to separate answers as needed. Type an integer or decimal rounded to four decimal places as needed.)
B.
There is no x-intercept.
yequals5xsquaredminus14xplus8
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your answer.
A.
The x-intercept(s) is/are xequals
enter your response here.
(Simplify your answer. Use a comma to separate answers as needed. Type an integer or decimal rounded to four decimal places as needed.)
B.
There is no x-intercept.
Answers
Answered by
GPT-4o mini
To find the x-intercepts of the function \( y = 5x^2 - 14x + 8 \), we can set \( y = 0 \) and solve for \( x \):
\[
5x^2 - 14x + 8 = 0
\]
This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \) where \( a = 5 \), \( b = -14 \), and \( c = 8 \). We can use the quadratic formula to find the x-intercepts:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substituting the values of \( a \), \( b \), and \( c \):
1. Calculate the discriminant \( b^2 - 4ac \):
\[
b^2 - 4ac = (-14)^2 - 4 \cdot 5 \cdot 8 = 196 - 160 = 36
\]
2. Now substituting back into the quadratic formula:
\[
x = \frac{14 \pm \sqrt{36}}{2 \cdot 5}
\]
\[
x = \frac{14 \pm 6}{10}
\]
3. This gives two possible solutions:
- First solution:
\[
x = \frac{14 + 6}{10} = \frac{20}{10} = 2
\]
- Second solution:
\[
x = \frac{14 - 6}{10} = \frac{8}{10} = 0.8
\]
Thus, the x-intercepts of the function are:
\[
x = 2 \quad \text{and} \quad x = 0.8
\]
### Conclusion
The correct choice is:
A. The x-intercept(s) is/are \( x = 2, 0.8 \) (approximate as needed).
B. There is no x-intercept is not applicable here since we have found two x-intercepts.
\[
5x^2 - 14x + 8 = 0
\]
This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \) where \( a = 5 \), \( b = -14 \), and \( c = 8 \). We can use the quadratic formula to find the x-intercepts:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substituting the values of \( a \), \( b \), and \( c \):
1. Calculate the discriminant \( b^2 - 4ac \):
\[
b^2 - 4ac = (-14)^2 - 4 \cdot 5 \cdot 8 = 196 - 160 = 36
\]
2. Now substituting back into the quadratic formula:
\[
x = \frac{14 \pm \sqrt{36}}{2 \cdot 5}
\]
\[
x = \frac{14 \pm 6}{10}
\]
3. This gives two possible solutions:
- First solution:
\[
x = \frac{14 + 6}{10} = \frac{20}{10} = 2
\]
- Second solution:
\[
x = \frac{14 - 6}{10} = \frac{8}{10} = 0.8
\]
Thus, the x-intercepts of the function are:
\[
x = 2 \quad \text{and} \quad x = 0.8
\]
### Conclusion
The correct choice is:
A. The x-intercept(s) is/are \( x = 2, 0.8 \) (approximate as needed).
B. There is no x-intercept is not applicable here since we have found two x-intercepts.