If x$^4$ + $\frac{1}{x^4}$ = 322, x ≠ 0, then one of the values of {x-$\frac{1}{x}$} is

Question

Accepted Answer

To solve the problem and justify why Option 2 is correct, we need to find the value of $ x - \frac{1}{x} $ given that $ x^4 + \frac{1}{x^4} = 322 $.

### Step-by-Step Explanation:

1. **Expression Simplification:**
   We start with the given equation:
   $$
   x^4 + \frac{1}{x^4} = 322
   $$

2. **Relation with $ x^2 + \frac{1}{x^2} $:**
   We know the identity:
   $$
   \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + 2 + \frac{1}{x^4}
   $$
   From this, we can express:
   $$
   x^4 + \frac{1}{x^4} = \left(x^2 + \frac{1}{x^2}\right)^2 - 2
   $$
   Substituting the given value:
   $$
   \left(x^2 + \frac{1}{x^2}\right)^2 - 2 = 322
   $$
   $$
   \left(x^2 + \frac{1}{x^2}\right)^2 = 324
   $$
   $$
   x^2 + \frac{1}{x^2} = \sqrt{324} = 18
   $$

3. **Finding $ x - \frac{1}{x} $:**
   We use another identity:
   $$
   \left(x - \frac{1}{x}\right)^2 = x^2 - 2 + \frac{1}{x^2}
   $$
   $$
   x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2
   $$
   Substituting the known value:
   $$
   18 = \left(x - \frac{1}{x}\right)^2 + 2
   $$
   $$
   \left(x - \frac{1}{x}\right)^2 = 16
   $$
   $$
   x - \frac{1}{x} = \sqrt{16} = 4
   $$

Thus, the value of $ x - \frac{1}{x} $ is 4, which corresponds to Option 2.

### Explanation of Incorrect Options:

- **Option 1 (2):** If $ x - \frac{1}{x} = 2 $, then:
  $$
  \left(x - \frac{1}{x}\right)^2 = 4
  $$
  $$
  x^2 + \frac{1}{x^2} = 4 + 2 = 6
  $$
  This does not satisfy the condition $ x^2 + \frac{1}{x^2} = 18 $.

- **Option 3 (6):** If $ x - \frac{1}{x} = 6 $, then:
  $$
  \left(x - \frac{1}{x}\right)^2 = 36
  $$
  $$
  x^2 + \frac{1}{x^2} = 36 + 2 = 38
  $$
  This also does not satisfy the condition $ x^2 + \frac{1}{x^2} = 18 $.

- **Option 4 (8):** If $ x - \frac{1}{x} = 8 $, then:
  $$
  \left(x - \frac{1}{x}\right)^2 = 64
  $$
  $$
  x^2 + \frac{1}{x^2} = 64 + 2 = 66
  $$
  This does not satisfy the condition $ x^2 + \frac{1}{x^2} = 18 $.

Therefore, Option 2 is the only correct answer, as it satisfies all the given conditions and identities.