If `p=(sqrt(q)+z)/(r^(2)+z)`, what is the value of z in terms of p, q, and r ?

Options

A. `(p-r^(2)sqrt(q))/(p-1)`
B. `(pr^(2)-sqrt(q))/(r^(2))`
C. `sqrt((p+qr)/(r^(2)))`
D. `(sqrt(q)-pr^(2))/(p-1)`

Explanation

To find the value of \( z \) in terms of \( p \), \( q \), and \( r \) from the equation

\[ p = \frac{\sqrt{q} + z}{r^2 + z} \]

we can follow these steps:

### Step 1: Cross-multiply the equation
We start by cross-multiplying to eliminate the fraction:

\[ p(r^2 + z) = \sqrt{q} + z \]

### Step 2: Distribute \( p \)
Next, we distribute \( p \) on the left side:

\[ pr^2 + pz = \sqrt{q} + z \]

### Step 3: Rearrange the equation
Now, we rearrange the equation to isolate \( z \):

\[ pz - z = \sqrt{q} - pr^2 \]

### Step 4: Factor out \( z \)
We can factor \( z \) out from the left side:

\[ z(p - 1) = \sqrt{q} - pr^2 \]

### Step 5: Solve for \( z \)
Finally, we solve for \( z \) by dividing both sides by \( (p - 1) \):

\[ z = \frac{\sqrt{q} - pr^2}{p - 1} \]

Thus, the value of \( z \) in terms of \( p \), \( q \), and \( r \) is:

\[ z = \frac{\sqrt{q} - pr^2}{p - 1} \]