If `x+1/x=p`, then find the value of `x^2+1/x^2`

Options

A. `P^2`
B. `P^2+2`
C. `P^2-2`
D. `P^2-4`

Answer

To find the value of \( x^2 + \frac{1}{x^2} \) given that \( x + \frac{1}{x} = p \), we can follow these steps:

### Step 1: Square the given equation
Start with the equation:
\[ x + \frac{1}{x} = p \]
Now, square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = p^2 \]

### Step 2: Expand the squared equation
Using the formula \( (A + B)^2 = A^2 + B^2 + 2AB \), we can expand the left side:
\[ x^2 + 2\left(x \cdot \frac{1}{x}\right) + \frac{1}{x^2} = p^2 \]
This simplifies to:
\[ x^2 + 2 + \frac{1}{x^2} = p^2 \]

### Step 3: Rearrange to find \( x^2 + \frac{1}{x^2} \)
Now, isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = p^2 - 2 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{p^2 - 2} \]