For the quadratic function f left parenthesis x right parenthesis equals negative 2 x squared plus 2 x minus 6​,        Determine where ​f(x)greater than0 and where ​f(x)less than0. Select the correct choice below and fill in the answer​ box(es) within your choice.
​(Type your answer in interval notation. Use integers or fractions for any numbers in the​ expression.)
A.
​f(x)greater than0 on 
  
enter your response here and ​f(x)less than0 on 
  
enter your response here
B.
​f(x)greater than0 on 
  
enter your response here and​ f(x) is never negative
C.
​f(x)less than0 on 
  
enter your response here and​ f(x) is never positive

To determine where \( f(x) > 0 \) and where \( f(x) < 0 \) for the given quadratic function

\[
f(x) = -2x^2 + 2x - 6
\]

we need to follow these steps:

### 1. Find the roots of the quadratic equation
Set \( f(x) = 0 \) to find where the function crosses the x-axis:

\[
-2x^2 + 2x - 6 = 0
\]

We can solve this by using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = -2 \), \( b = 2 \), and \( c = -6 \). Plugging in these values, we get:

\[
x = \frac{-2 \pm \sqrt{(2)^2 - 4(-2)(-6)}}{2(-2)}
\]
\[
x = \frac{-2 \pm \sqrt{4 - 48}}{-4}
\]
\[
x = \frac{-2 \pm \sqrt{-44}}{-4}
\]

Since the discriminant \((4 - 48)\) is negative, there are no real solutions. This means the parabola does not cross the x-axis, and therefore it either lies entirely above or entirely below the x-axis.

### 2. Determine the direction of the parabola
The coefficient of \( x^2 \) is \( -2 \), which is negative, meaning the parabola opens downward. 

Since the parabola opens downward and there are no real roots, the entire graph of the function lies below the x-axis, meaning:

\[
f(x) < 0 \quad \text{for all} \quad x \in (-\infty, \infty)
\]

### 3. Conclusion

Since \( f(x) \) is always negative, we choose:

**Choice C:**  
- \( f(x) < 0 \) on \( (-\infty, \infty) \)  
- \( f(x) \) is never positive.

---

Would you like further details or have any questions about this solution?

Here are 5 related questions to expand on this topic:
1. How does the sign of the leading coefficient affect the direction of a parabola?
2. What does it mean when the discriminant of a quadratic equation is negative?
3. How can we graphically interpret the results for a quadratic function with no real roots?
4. What are the key differences between a parabola that opens upward versus downward?
5. How can we identify whether a quadratic function has real roots without solving the equation?

**Tip:** When solving quadratic inequalities, the roots give critical points that split the number line into intervals. The sign of the function within each interval can then be tested.

For the quadratic function f(x) = -2x^2 + 2x - 6, determine where f(x) > 0 and where f(x) < 0. Select the correct choice below and fill in the answer box(es) within your choice. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

Learn how to solve quadratic function inequalities for f(x) = -2x² + 2x - 6, where f(x) > 0 and f(x) < 0. This problem involves finding the discriminant and understanding the nature of quadratic graphs. Ideal for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation