The value of 'c' for which `|alpha^(2) - beta^(2)| = 7//4`, where `alpha and beta` are the roots of `2 x^(2) + 7 x + c = 0`, is

Options

A. 4
B. 0
C. 6
D. 2

Answer

To solve the problem, we need to find the value of \( c \) for which \( |\alpha^2 - \beta^2| = \frac{7}{4} \), where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 2x^2 + 7x + c = 0 \).

### Step-by-Step Solution:

1. **Identify the roots**:
The roots \( \alpha \) and \( \beta \) of the quadratic equation \( 2x^2 + 7x + c = 0 \) can be expressed using Vieta's formulas:
- Sum of roots: \( \alpha + \beta = -\frac{b}{a} = -\frac{7}{2} \)
- Product of roots: \( \alpha \beta = \frac{c}{2} \)

2. **Use the identity for the difference of squares**:
We know that:
\[ |\alpha^2 - \beta^2| = |\alpha - \beta| \cdot |\alpha + \beta| \]
Given that \( |\alpha^2 - \beta^2| = \frac{7}{4} \), we can write:
\[ |\alpha - \beta| \cdot |\alpha + \beta| = \frac{7}{4} \]

3. **Substituting the known value**:
Substitute \( |\alpha + \beta| = \left| -\frac{7}{2} \right| = \frac{7}{2} \):
\[ |\alpha - \beta| \cdot \frac{7}{2} = \frac{7}{4} \]

4. **Solve for \( |\alpha - \beta| \)**:
Dividing both sides by \( \frac{7}{2} \):
\[ |\alpha - \beta| = \frac{7}{4} \cdot \frac{2}{7} = \frac{1}{2} \]

5. **Express \( |\alpha - \beta| \) in terms of \( c \)**:
We know that:
\[ |\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} \]
Substituting the values:
\[ |\alpha - \beta| = \sqrt{\left(-\frac{7}{2}\right)^2 - 4\left(\frac{c}{2}\right)} = \sqrt{\frac{49}{4} - 2c} \]

6. **Set up the equation**:
Now we can set the equation:
\[ \sqrt{\frac{49}{4} - 2c} = \frac{1}{2} \]

7. **Square both sides**:
Squaring both sides to eliminate the square root:
\[ \frac{49}{4} - 2c = \left(\frac{1}{2}\right)^2 \]
\[ \frac{49}{4} - 2c = \frac{1}{4} \]

8. **Rearranging the equation**:
Rearranging gives:
\[ 2c = \frac{49}{4} - \frac{1}{4} = \frac{48}{4} = 12 \]

9. **Solve for \( c \)**:
Dividing both sides by 2:
\[ c = \frac{12}{2} = 6 \]

### Final Answer:
The value of \( c \) is \( 6 \).