If the equation `(m+6)x^(2)+(m+6)x+2=0` has a pair of complex conjugate roots, then find interval of m

Options

A. `mgt-6`
B. `mlt2`
C. `-6ltmlt2`
D. `mlt-6ormgt2`

Answer

To determine the interval of \( m \) for which the equation \((m+6)x^2 + (m+6)x + 2 = 0\) has a pair of complex conjugate roots, we need to analyze the discriminant of the quadratic equation.

### Step 1: Identify coefficients
The given quadratic equation can be compared to the standard form \( ax^2 + bx + c = 0 \):
- \( a = m + 6 \)
- \( b = m + 6 \)
- \( c = 2 \)

### Step 2: Write the discriminant
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]
For the roots to be complex conjugates, the discriminant must be less than zero:
\[ D < 0 \]

### Step 3: Substitute the coefficients into the discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant:
\[ D = (m + 6)^2 - 4(m + 6)(2) \]
Simplifying this:
\[ D = (m + 6)^2 - 8(m + 6) \]

### Step 4: Factor the expression
Now, we can factor the expression:
\[ D = (m + 6)((m + 6) - 8) = (m + 6)(m - 2) \]
Thus, we have:
\[ (m + 6)(m - 2) < 0 \]

### Step 5: Solve the inequality
To solve the inequality \((m + 6)(m - 2) < 0\), we find the critical points where the expression equals zero:
- \( m + 6 = 0 \) gives \( m = -6 \)
- \( m - 2 = 0 \) gives \( m = 2 \)

### Step 6: Test intervals
We will test the sign of the product in the intervals determined by the critical points:
1. \( m < -6 \) (choose \( m = -7 \)):
\((m + 6)(m - 2) = (-1)(-9) > 0\)
2. \( -6 < m < 2 \) (choose \( m = 0 \)):
\((m + 6)(m - 2) = (6)(-2) < 0\)
3. \( m > 2 \) (choose \( m = 3 \)):
\((m + 6)(m - 2) = (9)(1) > 0\)

### Step 7: Conclusion
The inequality \((m + 6)(m - 2) < 0\) holds true in the interval:
\[ m \in (-6, 2) \]
Since we want the roots to be complex conjugates, we exclude the endpoints where the discriminant is zero.

### Final Answer
The interval of \( m \) for which the equation has a pair of complex conjugate roots is:
\[ m \in (-6, 2) \]