If `((1)/(16)p^(2)-q) =(1/4p-11) (1/4 p+11)` then q is

Options

A. 11
B. 1
C. 121
D. `11/4`

Answer

To solve the equation \(\frac{1}{16}p^2 - q = \left(\frac{1}{4}p - 11\right)\left(\frac{1}{4}p + 11\right)\), we will follow these steps:

### Step 1: Identify the equation
We start with the equation:
\[ \frac{1}{16}p^2 - q = \left(\frac{1}{4}p - 11\right)\left(\frac{1}{4}p + 11\right) \]

### Step 2: Apply the difference of squares formula
Recognize that the right-hand side can be simplified using the difference of squares formula, which states that \( (a - b)(a + b) = a^2 - b^2 \). Here, \( a = \frac{1}{4}p \) and \( b = 11 \).

So, we can rewrite the equation as:
\[ \frac{1}{16}p^2 - q = \left(\frac{1}{4}p\right)^2 - 11^2 \]

### Step 3: Calculate the squares
Now, calculate \( \left(\frac{1}{4}p\right)^2 \) and \( 11^2 \):
\[ \left(\frac{1}{4}p\right)^2 = \frac{1}{16}p^2 \]
\[ 11^2 = 121 \]

### Step 4: Substitute back into the equation
Substituting these values back into the equation gives:
\[ \frac{1}{16}p^2 - q = \frac{1}{16}p^2 - 121 \]

### Step 5: Isolate \( q \)
To isolate \( q \), we can rearrange the equation:
\[ -q = -121 \]
Thus, multiplying both sides by -1, we find:
\[ q = 121 \]

### Final Answer
The value of \( q \) is:
\[ \boxed{121} \]