For what values of k, the equation `x^2+2(k-4) x + 2k=0` has equal roots?

Options

A. 6 and 4
B. 8 and 2
C. 10 and 4
D. 12 and 2

Answer

To find the values of \( k \) for which the equation \( x^2 + 2(k-4)x + 2k = 0 \) has equal roots, we need to analyze the discriminant of the quadratic equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \).
Here, we have:
- \( a = 1 \)
- \( b = 2(k - 4) \)
- \( c = 2k \)

2. **Set up the discriminant condition**:
For a quadratic equation to have equal roots, the discriminant must be zero:
\[ D = b^2 - 4ac = 0 \]

3. **Substitute the coefficients into the discriminant**:
Substitute \( b \) and \( c \) into the discriminant formula:
\[ D = [2(k - 4)]^2 - 4(1)(2k) \]

4. **Expand the discriminant**:
Calculate \( D \):
\[ D = 4(k - 4)^2 - 8k \]
Expanding \( (k - 4)^2 \):
\[ D = 4(k^2 - 8k + 16) - 8k \]
\[ D = 4k^2 - 32k + 64 - 8k \]
\[ D = 4k^2 - 40k + 64 \]

5. **Set the discriminant to zero**:
Now, set the discriminant equal to zero:
\[ 4k^2 - 40k + 64 = 0 \]

6. **Simplify the equation**:
Divide the entire equation by 4 to simplify:
\[ k^2 - 10k + 16 = 0 \]

7. **Factor the quadratic equation**:
We need to factor \( k^2 - 10k + 16 \):
\[ (k - 2)(k - 8) = 0 \]

8. **Find the values of \( k \)**:
Set each factor to zero:
\[ k - 2 = 0 \quad \Rightarrow \quad k = 2 \]
\[ k - 8 = 0 \quad \Rightarrow \quad k = 8 \]

Thus, the values of \( k \) for which the equation has equal roots are \( k = 2 \) and \( k = 8 \).