If the zeroes of the quadratic `ax^(2)+bx+c` where `c ne 0`, are equal then:

Options

A. `c` and `a` have opposite signs
B. `c` and `b` have opposite signs
C. `c` and `a` have same signs
D. `c` and `b` have same signs

Explanation

To determine the conditions under which the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are equal, we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) is given by the formula:

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

### Step-by-Step Solution:

1. **Understand the Condition for Equal Roots**:
- For a quadratic equation to have equal roots, the discriminant must be zero.
- Therefore, we set the discriminant equal to zero:
\[ b^2 - 4ac = 0 \]

2. **Rearranging the Equation**:
- Rearranging the equation gives us:
\[ b^2 = 4ac \]

3. **Analyzing the Relationship**:
- Since \( c \neq 0 \) (as given in the problem), we can analyze the implications of \( b^2 = 4ac \).
- This implies that \( ac \) must be non-negative (i.e., \( ac \geq 0 \)) because \( b^2 \) is always non-negative.

4. **Identifying Cases for \( ac \geq 0 \)**:
- The product \( ac \geq 0 \) can occur in two scenarios:
- Both \( a \) and \( c \) are positive: \( a > 0 \) and \( c > 0 \).
- Both \( a \) and \( c \) are negative: \( a < 0 \) and \( c < 0 \).

5. **Conclusion**:
- From the analysis, we conclude that for the quadratic equation \( ax^2 + bx + c = 0 \) to have equal roots, \( a \) and \( c \) must have the same sign. Thus, the correct statement is that \( ac \geq 0 \).

### Summary of Results:
- The roots of the quadratic equation are equal if and only if \( ac \geq 0 \) (i.e., \( a \) and \( c \) have the same sign).