If $x + \frac{1}{x} = 6, then find x^2 + \frac{1}{x^2}$ :

Question

Accepted Answer

To solve the problem and justify why Option 3 is correct, we need to find the value of $ x^2 + \frac{1}{x^2} $ given that $ x + \frac{1}{x} = 6 $.

### Step-by-Step Explanation:

1. **Given Equation:**
   $$
   x + \frac{1}{x} = 6
   $$

2. **Square Both Sides:**
   To find $ x^2 + \frac{1}{x^2} $, we first square the given equation:
   $$
   \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}
   $$

3. **Expand the Squared Equation:**
   $$
   x^2 + 2 + \frac{1}{x^2} = 36
   $$
   This is because:
   $$
   (x + \frac{1}{x})^2 = 6^2 = 36
   $$

4. **Solve for $ x^2 + \frac{1}{x^2} $:**
   Subtract 2 from both sides to isolate $ x^2 + \frac{1}{x^2} $:
   $$
   x^2 + \frac{1}{x^2} = 36 - 2 = 34
   $$

Thus, the value of $ x^2 + \frac{1}{x^2} $ is 34, which corresponds to Option 3.

### Explanation of Why Other Options are Incorrect:

- **Option 1 (36):** This option is incorrect because it represents the value of $ (x + \frac{1}{x})^2 $, not $ x^2 + \frac{1}{x^2} $. After subtracting 2 from 36, we get 34, not 36.

- **Option 2 (30):** This option is incorrect because it does not match the calculated value of $ x^2 + \frac{1}{x^2} $. The correct calculation shows that the value is 34, not 30.

- **Option 4 (32):** This option is incorrect because it also does not match the calculated value. The subtraction of 2 from 36 gives 34, not 32.

In conclusion, the correct answer is Option 3, which is 34, as derived from the given equation and logical mathematical steps.