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 \( \frac{x}{2}-\frac{5}{x}=3 \).

To simplify \( \left(\frac{x}{2}-\frac{5}{x}\right)^{2} \), we start by applying the square of a binomial formula \( (a - b)^{2} = a^{2} - 2ab + b^{2} \). Here, \( a = \frac{x}{2} \) and \( b = \frac{5}{x} \). Calculating each term: - \( a^{2} = \left(\frac{x}{2}\right)^{2} = \frac{x^{2}}{4} \) - \( b^{2} = \left(\frac{5}{x}\right)^{2} = \frac{25}{x^{2}} \) - \( 2ab = 2\left(\frac{x}{2}\right)\left(\frac{5}{x}\right) = 5 \) Putting it all together, we have: \[ \left(\frac{x}{2}-\frac{5}{x}\right)^{2} = \frac{x^{2}}{4} - 5 + \frac{25}{x^{2}} \] Next, we need to calculate the value of \( \frac{x^{2}}{4}+\frac{25}{x^{2}} \) given \( \frac{x}{2}-\frac{5}{x}=3 \). Now, we can express \( \frac{x}{2} \) from the equation: \[ \frac{x}{2} = 3 + \frac{5}{x} \] Multiplying by \( x \) gives: \[ \frac{x^2}{2} = 3x + 5 \implies x^2 - 6x - 10 = 0 \] Using the quadratic formula to solve for \( x \): \[ x = \frac{6 \pm \sqrt{36 + 40}}{2} = 3 \pm \sqrt{19} \] Thus, for \( x = 3 + \sqrt{19} \) or \( x = 3 - \sqrt{19} \), substituting either back into \( \frac{x^{2}}{4} + \frac{25}{x^{2}} \) will yield the final values, which can be computed as follows based on our derived equations.