Determine rational numbers \(a\) and \(b\)
\[ \frac{3 – \sqrt{5}}{3 + 2\sqrt{5}} = a\sqrt{5} + b \]
Solution:
\[ \frac{3 – \sqrt{5}}{3 + 2\sqrt{5}} \times \frac{3 – 2\sqrt{5}}{3 – 2\sqrt{5}} \]
\[ = \frac{(3 – \sqrt{5})(3 – 2\sqrt{5})}{3^2 – (2\sqrt{5})^2} \]
\[ = \frac{9 – 6\sqrt{5} – 3\sqrt{5} + 2 \cdot 5}{9 – 20} \]
\[ = \frac{19 – 9\sqrt{5}}{-11} \]
\[ = -\frac{19}{11} + \frac{9}{11}\sqrt{5} \]
Comparing with \(a\sqrt{5} + b\)
\[ a = \frac{9}{11}, \quad b = -\frac{19}{11} \]