`f(x) = x^(2) + px + q`. If p = 2q and the function has a value of 16 at x = 3, what is the value of p ?

Answer

To solve the problem step by step, we will follow the given information and equations.

### Step 1: Write down the function and the given conditions
We have the function:
\[ f(x) = x^2 + px + q \]
We are given that:
1. \( p = 2q \)
2. \( f(3) = 16 \)

### Step 2: Substitute \( x = 3 \) into the function
We substitute \( x = 3 \) into the function:
\[ f(3) = 3^2 + p(3) + q \]

### Step 3: Simplify the equation
Calculating \( 3^2 \):
\[ f(3) = 9 + 3p + q \]

### Step 4: Set the equation equal to 16
Since we know that \( f(3) = 16 \), we can set up the equation:
\[ 9 + 3p + q = 16 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 3p + q = 16 - 9 \]
\[ 3p + q = 7 \]

### Step 6: Substitute \( q \) in terms of \( p \)
From the first condition \( p = 2q \), we can express \( q \) in terms of \( p \):
\[ q = \frac{p}{2} \]

### Step 7: Substitute \( q \) into the equation
Now substitute \( q \) into the equation \( 3p + q = 7 \):
\[ 3p + \frac{p}{2} = 7 \]

### Step 8: Combine the terms
To combine the terms, convert \( 3p \) into a fraction:
\[ 3p = \frac{6p}{2} \]
So the equation becomes:
\[ \frac{6p}{2} + \frac{p}{2} = 7 \]
\[ \frac{7p}{2} = 7 \]

### Step 9: Solve for \( p \)
To eliminate the fraction, multiply both sides by 2:
\[ 7p = 14 \]
Now divide by 7:
\[ p = 2 \]

### Conclusion
The value of \( p \) is:
\[ \boxed{2} \]