Simplify \[ \frac{5 - \sqrt{3}}{2 + \sqrt{3}} \], giving your answer in the form $a + b \sqrt{3}$, where $a$ and $b$ are integers. - Edexcel - A-Level Maths Pure - Question 5 - 2008 - Paper 2

The denominator can be simplified using the difference of squares:

[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1. ]

Now, we calculate the numerator:

[ (5 - \sqrt{3})(2 - \sqrt{3}) = 10 - 5\sqrt{3} - 2\sqrt{3} + 3 = 10 + 3 - 7\sqrt{3} = 13 - 7\sqrt{3}. ]

Since the denominator simplifies to 1, we can state that:

[ \frac{13 - 7\sqrt{3}}{1} = 13 - 7\sqrt{3}. ] Thus, in the form $a + b \sqrt{3}$, we have: $a = 13$ and $b = -7$.