If the root of the equation `Ax^2 - Bx + C = 0` are -1 and 1, then which one of the following is correct?

Options

A. A and Care both zero
B. A and B are both positive
C. A and C are both negative
D. A and C are of opposite sign

Answer

To solve the problem, we need to analyze the quadratic equation \( Ax^2 - Bx + C = 0 \) given that its roots are -1 and 1.

### Step-by-step Solution:

1. **Use Vieta's Formulas**: According to Vieta's formulas, for a quadratic equation \( Ax^2 + Bx + C = 0 \), if the roots are \( r_1 \) and \( r_2 \), then:
- The sum of the roots \( r_1 + r_2 = -\frac{B}{A} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{C}{A} \)

2. **Calculate the Sum of the Roots**:
- Here, the roots are -1 and 1.
- The sum of the roots is:
\[ -1 + 1 = 0 \]
- According to Vieta's, we have:
\[ -\frac{B}{A} = 0 \implies B = 0 \]

3. **Calculate the Product of the Roots**:
- The product of the roots is:
\[ (-1) \cdot 1 = -1 \]
- According to Vieta's, we have:
\[ \frac{C}{A} = -1 \implies C = -A \]

4. **Conclusion**:
- From the calculations, we found that \( B = 0 \) and \( C = -A \).
- This means that \( A \) and \( C \) are of opposite signs (if \( A \) is positive, \( C \) is negative, and vice versa).

### Final Result:
The correct conclusion is that \( A \) and \( C \) are of opposite signs.