If 3 – √3/2 + √3 = a + b√3, what are the values a and b?

The expression is the fraction (3 − √3)/(2 + √3). To remove the surd from the bottom, multiply the top and the bottom by the conjugate of the denominator, which is (2 − √3).

Bottom: (2 + √3)(2 − √3) = 2² − (√3)² = 4 − 3 = 1. Top: (3 − √3)(2 − √3) = 6 − 3√3 − 2√3 + (√3)² = 6 − 5√3 + 3 = 9 − 5√3.

So the fraction becomes (9 − 5√3)/1 = 9 − 5√3. Comparing with a + b√3, we get a = 9 and b = −5.