If $x+\frac{1}{x}=p, then x^2+\frac{1}{x^2}$ =

Question

Accepted Answer

To solve the problem and justify why Option 3 is correct, we need to find the expression for $x^2 + \frac{1}{x^2}$ given that $x + \frac{1}{x} = p$.

### Step-by-Step Explanation:

1. **Given Expression:**
   $$
   x + \frac{1}{x} = p
   $$

2. **Square Both Sides:**
   $$
   \left(x + \frac{1}{x}\right)^2 = p^2
   $$

3. **Expand the Squared Expression:**
   $$
   x^2 + 2 + \frac{1}{x^2} = p^2
   $$

4. **Rearrange to Find $x^2 + \frac{1}{x^2}$:**
   $$
   x^2 + \frac{1}{x^2} = p^2 - 2
   $$

Thus, the expression for $x^2 + \frac{1}{x^2}$ is $p^2 - 2$, which corresponds to Option 3.

### Explanation of Why Other Options Are Incorrect:

- **Option 1: $p^2$**
  - This option suggests that $x^2 + \frac{1}{x^2} = p^2$, which is incorrect because it does not account for the subtraction of 2 that arises from the expansion of $(x + \frac{1}{x})^2$.

- **Option 2: $p^2 + 2$**
  - This option incorrectly adds 2 instead of subtracting it. The correct derivation shows that 2 should be subtracted, not added.

- **Option 4: $p^2 - 4$**
  - This option incorrectly subtracts 4 instead of 2. The correct derivation shows that the subtraction should be 2, not 4.

In conclusion, the correct expression for $x^2 + \frac{1}{x^2}$ is $p^2 - 2$, which is why Option 3 is the correct answer.