If `4x ^(2) - 12 x + k ` is a perfect square, then the value of k is

Options

A. 2
B. 9
C. 12
D. 10

Answer

To determine the value of \( k \) such that the expression \( 4x^2 - 12x + k \) is a perfect square, we can follow these steps:

### Step 1: Identify the structure of a perfect square trinomial
A perfect square trinomial can be expressed in the form \( (ax + b)^2 \). For our expression \( 4x^2 - 12x + k \), we can rewrite it as \( (2x - 3)^2 \) plus some constant.

### Step 2: Expand the perfect square
Let's expand \( (2x - 3)^2 \):
\[ (2x - 3)^2 = 4x^2 - 12x + 9 \]

### Step 3: Compare with the original expression
Now we compare this with our original expression \( 4x^2 - 12x + k \):
\[ 4x^2 - 12x + k = 4x^2 - 12x + 9 \]

### Step 4: Set the constants equal
For the two expressions to be equal, the constant terms must also be equal:
\[ k = 9 \]

### Conclusion
Thus, the value of \( k \) that makes \( 4x^2 - 12x + k \) a perfect square is:
\[ \boxed{9} \]

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