The value of k for which the expression `9x ^(2) - kx + 16 ` be a perfect square is :

Options

A. `pm 12`
B. `pm 24`
C. `pm 2`
D. `pm 6`

Answer

To find the value of \( k \) for which the expression \( 9x^2 - kx + 16 \) is a perfect square, we can follow these steps:

### Step 1: Identify the standard form
The expression \( 9x^2 - kx + 16 \) can be compared to the standard quadratic form \( Ax^2 + Bx + C \), where:
- \( A = 9 \)
- \( B = -k \)
- \( C = 16 \)

### Step 2: Use the discriminant condition
For a quadratic expression to be a perfect square, the discriminant \( D \) must be equal to zero. The discriminant is given by the formula:
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = (-k)^2 - 4 \cdot 9 \cdot 16 \]
This simplifies to:
\[ D = k^2 - 576 \]

### Step 3: Set the discriminant to zero
Since we want the expression to be a perfect square, we set the discriminant \( D \) to zero:
\[ k^2 - 576 = 0 \]

### Step 4: Solve for \( k \)
Now, we solve the equation:
\[ k^2 = 576 \]
Taking the square root of both sides gives:
\[ k = \pm 24 \]

### Conclusion
Thus, the values of \( k \) for which the expression \( 9x^2 - kx + 16 \) is a perfect square are:
\[ k = 24 \quad \text{or} \quad k = -24 \]