Question

Given f of x is equal to 2 divided by the quantity x squared plus 3x minus 10 end quantity, which of the following is true?

f(x) is positive for all x < –5
f(x) is negative for all x < –5
f(x) is positive for all x < 2
f(x) is negative for all x > 2

Answer 1

To determine the sign of the function \( f(x) = \frac{2}{x^2 + 3x - 10} \), we first need to analyze the denominator \( x^2 + 3x - 10 \).

### Step 1: Find the roots of the quadratic equation
We can find the roots of the denominator by solving the equation:
\[
x^2 + 3x - 10 = 0
\]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 3, c = -10 \):
\[
x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1} = \frac{-3 \pm \sqrt{9 + 40}}{2} = \frac{-3 \pm \sqrt{49}}{2} = \frac{-3 \pm 7}{2}
\]
This gives us:
\[
x = \frac{4}{2} = 2 \quad \text{and} \quad x = \frac{-10}{2} = -5
\]
Thus, the roots are \( x = -5 \) and \( x = 2 \).

### Step 2: Analyze the sign of the denominator
Now we will analyze the sign of \( x^2 + 3x - 10 \) in the intervals defined by the roots \( x = -5 \) and \( x = 2 \).

1. **Interval \( (-\infty, -5) \)**:
- Choose \( x = -6 \):
\[
(-6)^2 + 3(-6) - 10 = 36 - 18 - 10 = 8 \quad (\text{positive})
\]

2. **Interval \( (-5, 2) \)**:
- Choose \( x = 0 \):
\[
0^2 + 3(0) - 10 = -10 \quad (\text{negative})
\]

3. **Interval \( (2, \infty) \)**:
- Choose \( x = 3 \):
\[
3^2 + 3(3) - 10 = 9 + 9 - 10 = 8 \quad (\text{positive})
\]

### Step 3: Determine the sign of \( f(x) \)
Now summarizing the signs of \( x^2 + 3x - 10 \):
- \( f(x) > 0 \) when \( x^2 + 3x - 10 > 0 \) which occurs in the intervals \( (-\infty, -5) \) and \( (2, \infty) \).
- \( f(x) < 0 \) when \( x^2 + 3x - 10 < 0 \) which occurs in the interval \( (-5, 2) \).

### Conclusion
Now, we can evaluate the statements provided:
- **\( f(x) \) is positive for all \( x < -5 \)** (True)
- **\( f(x) \) is negative for all \( x < -5 \)** (False)
- **\( f(x) \) is positive for all \( x < 2 \)** (False, it is negative between -5 and 2)
- **\( f(x) \) is negative for all \( x > 2 \)** (False, it is positive)

Thus, the correct answer is:
**\( f(x) \) is positive for all \( x < -5 \)**.