The equation `k^(2)x^(2)+kx+1=0` has

Options

A. one real root
B. two real roots
C. no real roots
D. None of these

Answer

To determine the nature of the roots of the quadratic equation \( k^2 x^2 + kx + 1 = 0 \), we can use the discriminant method. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:

\[ D = b^2 - 4ac \]

### Step-by-Step Solution:

1. **Identify coefficients**:
- From the equation \( k^2 x^2 + kx + 1 = 0 \), we can identify:
- \( a = k^2 \)
- \( b = k \)
- \( c = 1 \)

2. **Calculate the discriminant**:
- Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = b^2 - 4ac = k^2 - 4(k^2)(1) \]
- Simplifying this gives:
\[ D = k^2 - 4k^2 = k^2(1 - 4) = k^2(-3) \]
- Therefore, we have:
\[ D = -3k^2 \]

3. **Analyze the discriminant**:
- The discriminant \( D = -3k^2 \) is always less than 0 for any real value of \( k \) (since \( k^2 \) is always non-negative, and multiplying by -3 makes it negative).
- Since \( D < 0 \), this indicates that the quadratic equation has no real roots.

4. **Conclusion**:
- Therefore, the equation \( k^2 x^2 + kx + 1 = 0 \) has no real roots.

### Final Answer:
The equation has **no real roots**.

---