If $x + \frac{1}{x} = 4$, find the value of $x^2 + \frac{1}{x^2}$.

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**Solution:**

1. Given the equation:
   
   $$x + \frac{1}{x} = 4$$

2. Square both sides using the identity $(a + b)^2 = a^2 + 2ab + b^2$:

$$\left(x + \frac{1}{x}ight)^2 = 4^2$$

$$x^2 + 2 	imes x 	imes \frac{1}{x} + \frac{1}{x^2} = 16$$

3. Simplify the middle term:

$$x^2 + 2 + \frac{1}{x^2} = 16$$

4. Isolate $x^2 + \frac{1}{x^2}$:

$$x^2 + \frac{1}{x^2} = 16 - 2 = 14$$

**Answer:** $14$

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