If `n ne 3p`, and `s = (n^(2)+p)/(n-3p)`, what is the value of p in terms of n and s ?

Options

A. `(n^(2)-ns)/(1-3s)`
B. `(ns+n^(2))/(3s+1)`
C. `-(ns-n^(2))/(3s+1)`
D. `-(n^(2)-ns)/(3s+1)`

Explanation

To find the value of \( p \) in terms of \( n \) and \( s \) from the equation \( s = \frac{n^2 + p}{n - 3p} \), we can follow these steps:

### Step 1: Multiply both sides by the denominator
We start with the equation:
\[ s = \frac{n^2 + p}{n - 3p} \]
To eliminate the fraction, multiply both sides by \( n - 3p \):
\[ s(n - 3p) = n^2 + p \]

### Step 2: Distribute \( s \) on the left side
Distributing \( s \) gives us:
\[ sn - 3sp = n^2 + p \]

### Step 3: Rearrange the equation to isolate terms involving \( p \)
Now, we want to get all terms involving \( p \) on one side and the other terms on the opposite side:
\[ sn - n^2 = p + 3sp \]

### Step 4: Factor out \( p \)
We can factor \( p \) from the right side:
\[ sn - n^2 = p(1 + 3s) \]

### Step 5: Solve for \( p \)
Now, divide both sides by \( 1 + 3s \) to isolate \( p \):
\[ p = \frac{sn - n^2}{1 + 3s} \]

Thus, the value of \( p \) in terms of \( n \) and \( s \) is:
\[ \boxed{p = \frac{sn - n^2}{1 + 3s}} \]