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

Using the difference of squares:

$(2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1$

Expand the numerator:

$(5 - \sqrt{3})(2 - \sqrt{3}) = 10 - 5\sqrt{3} - 2\sqrt{3} + 3 = 13 - 7\sqrt{3}$

Thus we have:

$\frac{13 - 7\sqrt{3}}{1} = 13 - 7\sqrt{3}$

This can be expressed in the form $u + v\sqrt{3}$ with $a = 13$ and $b = -7$.