Find the value of k for which the roots of the quadratic equation `2x^(2) + kx + 8 = 0` will have equal value.

Answer

To find the value of \( k \) for which the roots of the quadratic equation \( 2x^2 + kx + 8 = 0 \) will have equal values, we need to use the condition for equal roots. The roots of a quadratic equation are equal when the discriminant \( D \) is equal to zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is \( 2x^2 + kx + 8 = 0 \). Here,
- \( a = 2 \) (coefficient of \( x^2 \))
- \( b = k \) (coefficient of \( x \))
- \( c = 8 \) (constant term)

2. **Write the formula for the discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]

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

4. **Simplify the expression**:
Calculate \( 4 \cdot 2 \cdot 8 \):
\[ 4 \cdot 2 = 8 \quad \text{and} \quad 8 \cdot 8 = 64 \]
So, the discriminant becomes:
\[ D = k^2 - 64 \]

5. **Set the discriminant equal to zero for equal roots**:
For the roots to be equal, set \( D = 0 \):
\[ k^2 - 64 = 0 \]

6. **Solve for \( k^2 \)**:
Rearranging gives:
\[ k^2 = 64 \]

7. **Take the square root of both sides**:
Taking the square root gives:
\[ k = \pm 8 \]

### Final Answer:
The values of \( k \) for which the roots of the quadratic equation will have equal values are:
\[ k = 8 \quad \text{or} \quad k = -8 \]