Find the nonzero value of k for which the roots of the quadratic equation `9x^(2)-3kx+k=0` are real and equal.

Answer

To find the non-zero value of \( k \) for which the roots of the quadratic equation \( 9x^2 - 3kx + k = 0 \) are real and equal, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we identify:
- \( a = 9 \)
- \( b = -3k \)
- \( c = k \)

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

### Step 3: Substitute the values of \( a \), \( b \), and \( c \)
Substituting the values we identified:
\[ (-3k)^2 - 4 \cdot 9 \cdot k = 0 \]

### Step 4: Simplify the equation
Calculating \( (-3k)^2 \):
\[ 9k^2 - 36k = 0 \]

### Step 5: Factor the equation
We can factor out \( 9k \):
\[ 9k(k - 4) = 0 \]

### Step 6: Solve for \( k \)
Setting each factor equal to zero gives us:
1. \( 9k = 0 \) → \( k = 0 \)
2. \( k - 4 = 0 \) → \( k = 4 \)

### Step 7: Identify the non-zero value
Since the question asks for the non-zero value of \( k \), we discard \( k = 0 \) and conclude:
\[ k = 4 \]

### Final Answer
The non-zero value of \( k \) for which the roots of the quadratic equation \( 9x^2 - 3kx + k = 0 \) are real and equal is:
\[ \boxed{4} \]