If the roots of the equation `x^(2)+8x-(2k+3)=0` differ by 2, then the value of k is

Options

A. 9
B. `-9`
C. 6
D. `-6`

Answer

To solve the problem, we need to find the value of \( k \) such that the roots of the quadratic equation \( x^2 + 8x - (2k + 3) = 0 \) differ by 2.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \).
Here, \( a = 1 \), \( b = 8 \), and \( c = -(2k + 3) \).

2. **Use the relationship of roots**:
Let the roots of the equation be \( \alpha \) and \( \beta \). The difference between the roots is given by:
\[ |\alpha - \beta| = 2 \]
The formula for the difference of the roots in terms of the discriminant \( D \) is:
\[ |\alpha - \beta| = \frac{\sqrt{D}}{a} \]
Since \( a = 1 \), we have:
\[ |\alpha - \beta| = \sqrt{D} \]
Therefore, we can set up the equation:
\[ \sqrt{D} = 2 \]

3. **Calculate the discriminant**:
The discriminant \( D \) for the quadratic equation is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = 8^2 - 4 \cdot 1 \cdot (-(2k + 3)) \]
Simplifying this:
\[ D = 64 + 4(2k + 3) \]
\[ D = 64 + 8k + 12 \]
\[ D = 76 + 8k \]

4. **Set up the equation**:
Now we substitute \( D \) back into the equation we derived from the roots:
\[ \sqrt{76 + 8k} = 2 \]

5. **Square both sides**:
Squaring both sides to eliminate the square root gives:
\[ 76 + 8k = 4 \]

6. **Solve for \( k \)**:
Rearranging the equation:
\[ 8k = 4 - 76 \]
\[ 8k = -72 \]
\[ k = \frac{-72}{8} = -9 \]

### Final Answer:
The value of \( k \) is \( -9 \).

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