To determine which option is wrong regarding the sets A and B, we need to analyze the quadratic equation given in the problem.
### Step 1: Identify the quadratic equation
The quadratic equation is given as:
\[ x^2 - (m + 1)x + (m + 4) = 0 \]
### Step 2: Determine the condition for real roots
For the roots of the quadratic equation to be real, the discriminant must be greater than or equal to zero. The discriminant \(D\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[ D = b^2 - 4ac \]
Here, \(a = 1\), \(b = -(m + 1)\), and \(c = m + 4\). Therefore, the discriminant is:
\[ D = (-(m + 1))^2 - 4 \cdot 1 \cdot (m + 4) \]
\[ D = (m + 1)^2 - 4(m + 4) \]
### Step 3: Expand the discriminant
Expanding the expression:
\[ D = (m^2 + 2m + 1) - (4m + 16) \]
\[ D = m^2 + 2m + 1 - 4m - 16 \]
\[ D = m^2 - 2m - 15 \]
### Step 4: Set the discriminant greater than or equal to zero
To find the values of \(m\) for which the roots are real:
\[ m^2 - 2m - 15 \geq 0 \]
### Step 5: Factor the quadratic
Factoring the quadratic:
\[ (m - 5)(m + 3) \geq 0 \]
### Step 6: Determine the intervals
To solve the inequality, we find the roots:
- \(m - 5 = 0 \Rightarrow m = 5\)
- \(m + 3 = 0 \Rightarrow m = -3\)
Now we test the intervals:
1. \(m < -3\)
2. \(-3 \leq m \leq 5\)
3. \(m > 5\)
Using test points:
- For \(m < -3\) (e.g., \(m = -4\)): \(((-4) - 5)((-4) + 3) = (-9)(-1) > 0\) (True)
- For \(-3 < m < 5\) (e.g., \(m = 0\)): \((0 - 5)(0 + 3) = (-5)(3) < 0\) (False)
- For \(m > 5\) (e.g., \(m = 6\)): \((6 - 5)(6 + 3) = (1)(9) > 0\) (True)
### Step 7: Conclusion for set A
Thus, the solution for the inequality is:
\[ m \in (-\infty, -3] \cup [5, \infty) \]
### Step 8: Define set B
Set B is given as:
\[ B = [-3, 5) \]
### Step 9: Analyze the options
Now we need to check the options provided to see which one is wrong:
1. \(A - B\)
2. \(A \cap B\)
3. \(A \cup B\)
4. \(B - A\)
### Step 10: Calculate A - B
- \(A - B = (-\infty, -3] \cup [5, \infty) - [-3, 5)\)
- This results in:
\[ (-\infty, -3) \cup [5, \infty) \]
### Step 11: Calculate A ∩ B
- \(A \cap B = (-\infty, -3] \cap [-3, 5) = \{-3\}\)
### Step 12: Calculate A ∪ B
- \(A \cup B = (-\infty, -3] \cup [-3, 5) \cup [5, \infty) = (-\infty, \infty)\)
### Step 13: Calculate B - A
- \(B - A = [-3, 5) - ((-\infty, -3] \cup [5, \infty))\)
- This results in:
\[ (-3, 5) \]
### Step 14: Identify the wrong option
From the calculations:
- \(A - B\) is correct.
- \(A \cap B\) is correct.
- \(A \cup B\) is correct.
- \(B - A\) is correct.
The wrong option is the one that incorrectly represents the intersection or union based on our calculations.
### Final Answer
The wrong option is the one that misrepresents the intersection or union of sets A and B.
