If `x + (1)/(2x) = 4`, then find `x^(2) + (1)/(4x^(2))` .

Answer

To solve the equation \( x + \frac{1}{2x} = 4 \) and find the value of \( x^2 + \frac{1}{4x^2} \), we will follow these steps:

### Step 1: Square both sides of the equation
Start with the given equation:
\[ x + \frac{1}{2x} = 4 \]
Now, square both sides:
\[ \left(x + \frac{1}{2x}\right)^2 = 4^2 \]

### Step 2: Apply the square of a binomial formula
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \), where \( a = x \) and \( b = \frac{1}{2x} \):
\[ x^2 + 2\left(x \cdot \frac{1}{2x}\right) + \left(\frac{1}{2x}\right)^2 = 16 \]

### Step 3: Simplify the equation
Calculate \( 2\left(x \cdot \frac{1}{2x}\right) \) which simplifies to \( 1 \):
\[ x^2 + 1 + \left(\frac{1}{2x}\right)^2 = 16 \]
Next, calculate \( \left(\frac{1}{2x}\right)^2 \):
\[ \left(\frac{1}{2x}\right)^2 = \frac{1}{4x^2} \]
So the equation becomes:
\[ x^2 + 1 + \frac{1}{4x^2} = 16 \]

### Step 4: Rearrange the equation
Now, isolate \( x^2 + \frac{1}{4x^2} \):
\[ x^2 + \frac{1}{4x^2} = 16 - 1 \]
\[ x^2 + \frac{1}{4x^2} = 15 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{4x^2} \) is:
\[ \boxed{15} \]