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 \( \frac{x}{2}-\frac{5}{x}=3 \). 3. Complete the following. a) Simplify \( \left(\frac{x}{3}-\frac{6}{x}\right)^{2} \). b) Then evaluate \( \frac{x^{2}}{9}+\frac{36}{x^{2}} \), if \( \frac{x}{3}-\frac{6}{x}=1 \).

Sure, let's solve each part step by step. --- ## Problem 2 ### 2.a) Simplify \( \left(\frac{x}{2} - \frac{5}{x}\right)^{2} \) **Solution:** We can use the algebraic identity for squaring a binomial: \[ (a - b)^2 = a^2 - 2ab + b^2 \] Let \( a = \frac{x}{2} \) and \( b = \frac{5}{x} \). Applying the identity: \[ \left(\frac{x}{2} - \frac{5}{x}\right)^2 = \left(\frac{x}{2}\right)^2 - 2 \cdot \frac{x}{2} \cdot \frac{5}{x} + \left(\frac{5}{x}\right)^2 \] Now, calculate each term: 1. **First Term:** \[ \left(\frac{x}{2}\right)^2 = \frac{x^2}{4} \] 2. **Second Term:** \[ 2 \cdot \frac{x}{2} \cdot \frac{5}{x} = 2 \cdot \frac{5}{2} = 5 \] Since it's subtracted in the original identity: \[ -2ab = -5 \] 3. **Third Term:** \[ \left(\frac{5}{x}\right)^2 = \frac{25}{x^2} \] **Combined Simplification:** \[ \left(\frac{x}{2} - \frac{5}{x}\right)^2 = \frac{x^2}{4} - 5 + \frac{25}{x^2} \] --- ### 2.b) Calculate the value of \( \frac{x^{2}}{4} + \frac{25}{x^{2}} \), if \( \frac{x}{2} - \frac{5}{x} = 3 \) **Solution:** From part 2.a, we have: \[ \left(\frac{x}{2} - \frac{5}{x}\right)^2 = \frac{x^2}{4} + \frac{25}{x^2} - 5 \] Given that: \[ \frac{x}{2} - \frac{5}{x} = 3 \] Square both sides: \[ \left(\frac{x}{2} - \frac{5}{x}\right)^2 = 3^2 = 9 \] Substitute back: \[ \frac{x^2}{4} + \frac{25}{x^2} - 5 = 9 \] Solve for \( \frac{x^2}{4} + \frac{25}{x^2} \): \[ \frac{x^2}{4} + \frac{25}{x^2} = 9 + 5 = 14 \] **Answer:** \( \frac{x^{2}}{4} + \frac{25}{x^{2}} = 14 \) --- ## Problem 3 ### 3.a) Simplify \( \left(\frac{x}{3} - \frac{6}{x}\right)^{2} \) **Solution:** Again, use the binomial square identity: \[ (a - b)^2 = a^2 - 2ab + b^2 \] Let \( a = \frac{x}{3} \) and \( b = \frac{6}{x} \). Applying the identity: \[ \left(\frac{x}{3} - \frac{6}{x}\right)^2 = \left(\frac{x}{3}\right)^2 - 2 \cdot \frac{x}{3} \cdot \frac{6}{x} + \left(\frac{6}{x}\right)^2 \] Now, calculate each term: 1. **First Term:** \[ \left(\frac{x}{3}\right)^2 = \frac{x^2}{9} \] 2. **Second Term:** \[ 2 \cdot \frac{x}{3} \cdot \frac{6}{x} = 2 \cdot \frac{6}{3} = 2 \cdot 2 = 4 \] With the negative sign from the identity: \[ -2ab = -4 \] 3. **Third Term:** \[ \left(\frac{6}{x}\right)^2 = \frac{36}{x^2} \] **Combined Simplification:** \[ \left(\frac{x}{3} - \frac{6}{x}\right)^2 = \frac{x^2}{9} - 4 + \frac{36}{x^2} \] --- ### 3.b) Evaluate \( \frac{x^{2}}{9} + \frac{36}{x^{2}} \), if \( \frac{x}{3} - \frac{6}{x} = 1 \) **Solution:** From part 3.a, we have: \[ \left(\frac{x}{3} - \frac{6}{x}\right)^2 = \frac{x^2}{9} + \frac{36}{x^2} - 4 \] Given that: \[ \frac{x}{3} - \frac{6}{x} = 1 \] Square both sides: \[ \left(\frac{x}{3} - \frac{6}{x}\right)^2 = 1^2 = 1 \] Substitute back: \[ \frac{x^2}{9} + \frac{36}{x^2} - 4 = 1 \] Solve for \( \frac{x^2}{9} + \frac{36}{x^2} \): \[ \frac{x^2}{9} + \frac{36}{x^2} = 1 + 4 = 5 \] **Answer:** \( \frac{x^{2}}{9} + \frac{36}{x^{2}} = 5 \) ---