Find the value correct to three decimal places
\[ \frac{3 – \sqrt{5}}{3 + 2\sqrt{5}}, \quad \text{where } \sqrt{5} = 2.2360 \]
Solution:
\[ \frac{3 – \sqrt{5}}{3 + 2\sqrt{5}} \times \frac{3 – 2\sqrt{5}}{3 – 2\sqrt{5}} \]
\[ = \frac{(3 – \sqrt{5})(3 – 2\sqrt{5})}{9 – 20} \]
\[ = \frac{9 – 6\sqrt{5} – 3\sqrt{5} + 10}{-11} \]
\[ = \frac{19 – 9\sqrt{5}}{-11} \]
\[ = -\frac{19}{11} + \frac{9}{11}\sqrt{5} \]
\[ = -1.727 + \frac{9}{11}(2.2360) \]
\[ = -1.727 + 1.829 \]
\[ = 0.102 \ (\text{correct to three decimal places}) \]