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 the quadratic equation \( 2x^2 + 8x + k = 0 \) such that the roots are equal, we follow these steps:

### Step 1: Understand the condition for equal roots
For a quadratic equation \( ax^2 + bx + c = 0 \), the roots are equal when the discriminant \( D \) is equal to zero. The discriminant is given by the formula:
\[ D = b^2 - 4ac \]

### Step 2: Identify coefficients
In the given equation \( 2x^2 + 8x + k = 0 \), we can identify the coefficients:
- \( a = 2 \)
- \( b = 8 \)
- \( c = k \)

### Step 3: Set up the discriminant equation
Since we want the roots to be equal, we set the discriminant to zero:
\[ D = b^2 - 4ac = 0 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ 8^2 - 4 \cdot 2 \cdot k = 0 \]

### Step 4: Simplify the equation
Calculating \( 8^2 \):
\[ 64 - 8k = 0 \]

### Step 5: Solve for \( k \)
Rearranging the equation gives:
\[ 64 = 8k \]
Now, divide both sides by 8:
\[ k = \frac{64}{8} = 8 \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{8} \]

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