If the difference between the roots of the equation ` Ax^(2) - B x + C = 0 ` is 4, then which of the following is TURE ?

Options

A. `B^(2) - 16 a ^(2) = 4AC + 4 B^(2)`
B. ` B^(2) - 10 A^(2) = 4AC + 6A^(2)`
C. `B^(2) - 8 A^(2) = 4AC + 10 A^(2)`
D. `B^(2) - 16 A^(2) = 4AC + 8 B^(2)`

Answer

To solve the problem, we need to analyze the quadratic equation \( Ax^2 - Bx + C = 0 \) and the condition that the difference between its roots is 4.

### Step-by-step Solution:

1. **Identify the Roots**:
For the quadratic equation \( Ax^2 - Bx + C = 0 \), the roots can be denoted as \( \alpha \) and \( \beta \).

2. **Sum and Product of Roots**:
From Vieta's formulas:
- The sum of the roots \( \alpha + \beta = \frac{B}{A} \)
- The product of the roots \( \alpha \beta = \frac{C}{A} \)

3. **Difference of Roots**:
The difference between the roots is given by:
\[ \alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} \]
Substituting the values from Vieta's formulas:
\[ \alpha - \beta = \sqrt{\left(\frac{B}{A}\right)^2 - 4\left(\frac{C}{A}\right)} = \sqrt{\frac{B^2}{A^2} - \frac{4C}{A}} \]
This simplifies to:
\[ \alpha - \beta = \sqrt{\frac{B^2 - 4AC}{A^2}} \]

4. **Setting the Difference Equal to 4**:
Given that the difference between the roots is 4, we set up the equation:
\[ \sqrt{\frac{B^2 - 4AC}{A^2}} = 4 \]

5. **Squaring Both Sides**:
Squaring both sides to eliminate the square root gives:
\[ \frac{B^2 - 4AC}{A^2} = 16 \]

6. **Rearranging the Equation**:
Multiplying both sides by \( A^2 \) results in:
\[ B^2 - 4AC = 16A^2 \]

7. **Final Form**:
Rearranging gives us:
\[ B^2 - 4AC - 16A^2 = 0 \]

### Conclusion:
From the above steps, we conclude that the condition \( B^2 - 4AC = 16A^2 \) must hold true.