Find the values of \(x\) and \(y\)
If \[ \frac{5 – \sqrt{3}}{2 + \sqrt{3}} = x + y\sqrt{3} \]
(a) \(x = 13, y = -7\) \quad (b) \(x = -13, y = 7\) \quad (c) \(x = -13, y = -7\) \quad (d) \(x = 13, y = 7\)
Solution:
\[ \frac{5 – \sqrt{3}}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} \]
\[ = \frac{(5 – \sqrt{3})(2 – \sqrt{3})}{4 – 3} = 13 – 7\sqrt{3} \]
\[ \therefore x = 13,\quad y = -7 \ (\text{Option a}) \]