Simplify (5-2√3)/(√3-1) giving your answer in the form p +q√3, where p and q are rational numbers

The trick here is to use a technique called the difference of squares. If we multiply the top and bottom of the fraction by the conjugate* of the denominator, we can remove any square root terms from the denominator.

*If the denominator is √3-1, its conjugate is √3+1.

((5-2√3)(√3+1))/((√3-1)(√3+1)) = (5√3 - 2√3 - 6 + 5)/(3 - √3 + √3 -1) = (3√3-1)/2= (3/2)*√3 -1/2