By using an appropriate identity, expand the following:
`(x/2 - 2/(x))^(2)`

Answer

To expand the expression \((\frac{x}{2} - \frac{2}{x})^{2}\), we can use the identity for the square of a binomial, which states that:

\[ (a - b)^{2} = a^{2} - 2ab + b^{2} \]

### Step 1: Identify \(a\) and \(b\)

In our case, we can identify:
- \(a = \frac{x}{2}\)
- \(b = \frac{2}{x}\)

### Step 2: Calculate \(a^{2}\)

Now, we calculate \(a^{2}\):

\[ a^{2} = \left(\frac{x}{2}\right)^{2} = \frac{x^{2}}{4} \]

### Step 3: Calculate \(b^{2}\)

Next, we calculate \(b^{2}\):

\[ b^{2} = \left(\frac{2}{x}\right)^{2} = \frac{4}{x^{2}} \]

### Step 4: Calculate \(2ab\)

Now we calculate \(2ab\):

\[ 2ab = 2 \cdot \left(\frac{x}{2}\right) \cdot \left(\frac{2}{x}\right) = 2 \cdot \frac{x \cdot 2}{2x} = 2 \]

### Step 5: Substitute into the identity

Now we substitute \(a^{2}\), \(b^{2}\), and \(2ab\) into the binomial expansion formula:

\[ (\frac{x}{2} - \frac{2}{x})^{2} = a^{2} - 2ab + b^{2} \]

Substituting the values we calculated:

\[ = \frac{x^{2}}{4} - 2 + \frac{4}{x^{2}} \]

### Step 6: Combine the terms

Thus, the expanded form of the expression is:

\[ \frac{x^{2}}{4} - 2 + \frac{4}{x^{2}} \]

### Final Answer

The final expanded expression is:

\[ \frac{x^{2}}{4} - 2 + \frac{4}{x^{2}} \]

---