To solve the problem, we need to analyze the quadratic equation given as \( x^2 + nx + m = 0 \) where \( n \) and \( m \) are integers. We want to determine which type of roots this equation cannot have.
### Step-by-Step Solution:
1. **Identify the standard form of the quadratic equation**:
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). In our case, we have:
\[
a = 1, \quad b = n, \quad c = m
\]
2. **Use the quadratic formula to find the roots**:
The roots of the quadratic equation are given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substituting \( a \), \( b \), and \( c \) into the formula, we get:
\[
x = \frac{-n \pm \sqrt{n^2 - 4m}}{2}
\]
3. **Analyze the discriminant**:
The term under the square root, \( n^2 - 4m \), is called the discriminant. We will analyze its value to determine the nature of the roots:
- If \( n^2 - 4m > 0 \): The roots are real and distinct.
- If \( n^2 - 4m = 0 \): The roots are real and equal (integral roots).
- If \( n^2 - 4m < 0 \): The roots are complex.
4. **Check the conditions for integral, non-integral, irrational, and complex roots**:
- **Integral roots**: Occur when \( n^2 - 4m \) is a perfect square.
- **Non-integral roots**: Occur when \( n^2 - 4m \) is positive but not a perfect square.
- **Irrational roots**: Occur when \( n^2 - 4m \) is positive and not a perfect square.
- **Complex roots**: Occur when \( n^2 - 4m < 0 \).
5. **Determine which type of roots cannot occur**:
- Since \( n \) and \( m \) are integers, if \( n^2 - 4m < 0 \), the roots will be complex.
- If \( n^2 - 4m = 0 \), the roots will be integral.
- If \( n^2 - 4m > 0 \), we can have both integral and non-integral (including irrational) roots depending on whether \( n^2 - 4m \) is a perfect square.
6. **Conclusion**:
The equation \( x^2 + nx + m = 0 \) cannot have **complex roots** when \( n \) and \( m \) are integers, as it can have integral, non-integral, and irrational roots depending on the discriminant.
### Final Answer:
The equation \( x^2 + nx + m = 0 \) cannot have **complex roots**.
