3. Given: \( \quad x+\frac{5}{x}=6 \) Determine the value of \( x^{2}+\frac{25}{x^{2}} \)

To find the value of \( x^{2} + \frac{25}{x^{2}} \) given the equation \( x + \frac{5}{x} = 6 \), we can follow these steps: 1. **Square both sides of the given equation**: \[ \left( x + \frac{5}{x} \right)^{2} = 6^{2} \] 2. **Expand the left side**: \[ x^{2} + 2 \cdot x \cdot \frac{5}{x} + \frac{25}{x^{2}} = 36 \] This simplifies to: \[ x^{2} + 10 + \frac{25}{x^{2}} = 36 \] 3. **Rearrange the equation**: \[ x^{2} + \frac{25}{x^{2}} = 36 - 10 \] \[ x^{2} + \frac{25}{x^{2}} = 26 \] Thus, the value of \( x^{2} + \frac{25}{x^{2}} \) is \( \boxed{26} \).