2. Complete the following. a) Simplify \( \left(\frac{x}{2}-\frac{5}{x}\right)^{2} \). b) Calculate the value of \( \frac{x^{2}}{4}+\frac{25}{x^{2}} \), If \( \frac{x}{2}-\frac{5}{x}=3 \).

**a)** We start with \[ \left(\frac{x}{2} - \frac{5}{x}\right)^2. \] By applying the formula \((a - b)^2 = a^2 - 2ab + b^2\), with \( a = \frac{x}{2} \) and \( b = \frac{5}{x} \), we have \[ \left(\frac{x}{2} - \frac{5}{x}\right)^2 = \left(\frac{x}{2}\right)^2 - 2\left(\frac{x}{2}\right)\left(\frac{5}{x}\right) + \left(\frac{5}{x}\right)^2. \] 1. Compute the first term: \[ \left(\frac{x}{2}\right)^2 = \frac{x^2}{4}. \] 2. Compute the middle term: \[ 2\left(\frac{x}{2}\right)\left(\frac{5}{x}\right) = 2 \cdot \frac{5}{2} = 5. \] Since it is subtracted, we have \(-5\). 3. Compute the last term: \[ \left(\frac{5}{x}\right)^2 = \frac{25}{x^2}. \] Thus, the simplified form is \[ \frac{x^2}{4} - 5 + \frac{25}{x^2}. \] --- **b)** We are given \[ \frac{x}{2} - \frac{5}{x} = 3. \] Square both sides: \[ \left(\frac{x}{2} - \frac{5}{x}\right)^2 = 3^2. \] From part (a) we already found that \[ \left(\frac{x}{2} - \frac{5}{x}\right)^2 = \frac{x^2}{4} - 5 + \frac{25}{x^2}. \] So, \[ \frac{x^2}{4} - 5 + \frac{25}{x^2} = 9. \] To find the value of \(\frac{x^2}{4} + \frac{25}{x^2}\), add 5 to both sides: \[ \frac{x^2}{4} + \frac{25}{x^2} = 9 + 5 = 14. \] --- \(\boxed{\text{a) } \frac{x^2}{4} - 5 + \frac{25}{x^2}, \quad \text{b) } 14}\)