If the roots of the quadratic equation `2x^(2)+8x+k=0` are equal, find the value of `k`.

Options

A. `k=0`
B. `k=4`
C. `k=8`
D. `k=2`

Answer

To find the value of \( k \) for which the roots of the quadratic equation \( 2x^2 + 8x + k = 0 \) are equal, we will use the condition that the discriminant of the quadratic equation must be zero.

### Step-by-Step Solution:

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

2. **Write the formula for the discriminant**:
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]

3. **Substitute the values into the discriminant formula**:
Plugging in the values of \( a \), \( b \), and \( c \):
\[ D = 8^2 - 4 \cdot 2 \cdot k \]

4. **Simplify the expression**:
Calculate \( 8^2 \):
\[ D = 64 - 8k \]

5. **Set the discriminant to zero for equal roots**:
Since the roots are equal, we set the discriminant \( D \) to zero:
\[ 64 - 8k = 0 \]

6. **Solve for \( k \)**:
Rearranging the equation gives:
\[ 8k = 64 \]
Dividing both sides by 8:
\[ k = \frac{64}{8} = 8 \]

### Final Answer:
The value of \( k \) is \( 8 \).

---