`sqrt(3p^(2)-11)-x=0`
If `pgt0 and x=8` in the equation above, what is the value of p?

Answer

To solve the equation \( \sqrt{3p^2 - 11} - x = 0 \) given that \( p > 0 \) and \( x = 8 \), we can follow these steps:

### Step 1: Substitute the value of \( x \)
We start by substituting \( x = 8 \) into the equation:
\[ \sqrt{3p^2 - 11} - 8 = 0 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to isolate the square root:
\[ \sqrt{3p^2 - 11} = 8 \]

### Step 3: Square both sides
Now we square both sides to eliminate the square root:
\[ (\sqrt{3p^2 - 11})^2 = 8^2 \]
This simplifies to:
\[ 3p^2 - 11 = 64 \]

### Step 4: Solve for \( 3p^2 \)
Next, we add 11 to both sides to solve for \( 3p^2 \):
\[ 3p^2 = 64 + 11 \]
\[ 3p^2 = 75 \]

### Step 5: Divide by 3
Now, we divide both sides by 3 to find \( p^2 \):
\[ p^2 = \frac{75}{3} \]
\[ p^2 = 25 \]

### Step 6: Take the square root
Taking the square root of both sides gives us:
\[ p = \pm 5 \]

### Step 7: Consider the condition \( p > 0 \)
Since we know that \( p > 0 \), we discard the negative solution:
\[ p = 5 \]

Thus, the value of \( p \) is \( \boxed{5} \).

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