Find the square root of :
`4 a ^(2) + 9b ^(2) + c ^(2) - 12 ab + 6 bc - 4 ac`

Options

A. `(2 a + 3 b - c)`
B. `(2a - 3 b + c)`
C. `(-2a + 3b + c)`
D. `(-2a + 3b + c)`

Answer

To find the square root of the expression \(4a^2 + 9b^2 + c^2 - 12ab + 6bc - 4ac\), we can follow these steps:

### Step 1: Group the terms
We start with the expression:
\[ 4a^2 + 9b^2 + c^2 - 12ab + 6bc - 4ac \]

### Step 2: Recognize the perfect square structure
Notice that the expression can be rearranged to fit the form of a perfect square trinomial. We can rewrite \(4a^2\) as \((2a)^2\), \(9b^2\) as \((3b)^2\), and \(c^2\) as \((c)^2\).

### Step 3: Identify the terms
We can express the terms as follows:
- \(4a^2 = (2a)^2\)
- \(9b^2 = (3b)^2\)
- \(c^2 = (c)^2\)

### Step 4: Write the expression in a square form
Now, we can rewrite the expression:
\[ (2a)^2 + (3b)^2 + (c)^2 - 12ab + 6bc - 4ac \]

### Step 5: Use the identity for squares
We can use the identity \((x - y - z)^2 = x^2 + y^2 + z^2 - 2xy - 2xz - 2yz\) to factor the expression. Here, we can let:
- \(x = 2a\)
- \(y = 3b\)
- \(z = c\)

Then, we can rewrite the expression as:
\[ (2a - 3b - c)^2 \]

### Step 6: Take the square root
Now, we can take the square root of the entire expression:
\[ \sqrt{(2a - 3b - c)^2} = |2a - 3b - c| \]

### Final Answer
Thus, the square root of the expression \(4a^2 + 9b^2 + c^2 - 12ab + 6bc - 4ac\) is:
\[ |2a - 3b - c| \]