Find the value(s) of `k` so that, the quadratic equation `x^2 - 4kx + k = 0` has equal roots.

Answer

To find the value(s) of \( k \) such that the quadratic equation \( x^2 - 4kx + k = 0 \) has equal roots, we follow these steps:

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

### Step 2: Use the condition for equal roots
For a quadratic equation to have equal roots, the discriminant must be equal to zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Setting the discriminant to zero:
\[ D = 0 \implies (-4k)^2 - 4 \cdot 1 \cdot k = 0 \]

### Step 3: Simplify the equation
Now, we simplify the equation:
\[ 16k^2 - 4k = 0 \]

### Step 4: Factor the equation
We can factor out \( 4k \):
\[ 4k(4k - 1) = 0 \]

### Step 5: Solve for \( k \)
Setting each factor equal to zero gives us:
1. \( 4k = 0 \) which implies \( k = 0 \)
2. \( 4k - 1 = 0 \) which implies \( 4k = 1 \) or \( k = \frac{1}{4} \)

### Conclusion
Thus, the values of \( k \) for which the quadratic equation has equal roots are:
\[ k = 0 \quad \text{or} \quad k = \frac{1}{4} \]