Determine k for which the roots of the equation `9x^(2)+2kx+4=0` are equal.

Answer

To determine the value of \( k \) for which the roots of the equation \( 9x^2 + 2kx + 4 = 0 \) are equal, we need to follow these steps:

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

### Step 2: Identify coefficients
In our equation \( 9x^2 + 2kx + 4 = 0 \):
- \( a = 9 \)
- \( b = 2k \)
- \( c = 4 \)

### Step 3: Write the discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (2k)^2 - 4 \cdot 9 \cdot 4 \]

### Step 4: Simplify the discriminant
Calculating \( D \):
\[ D = 4k^2 - 144 \]

### Step 5: Set the discriminant to zero
For the roots to be equal, we set the discriminant to zero:
\[ 4k^2 - 144 = 0 \]

### Step 6: Solve for \( k^2 \)
Rearranging the equation gives:
\[ 4k^2 = 144 \]
Dividing both sides by 4:
\[ k^2 = 36 \]

### Step 7: Solve for \( k \)
Taking the square root of both sides:
\[ k = \pm 6 \]

### Final Answer
Thus, the values of \( k \) for which the roots of the equation are equal are:
\[ k = 6 \quad \text{or} \quad k = -6 \]

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