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Find rational numbers a and b such that (2 + 3 sqrt(5)) / (1 - 3 sqrt(5)) = a + b sqrt(5).

1. a = -47/44, b = -9/44
2. a = 47/44, b = -9/44
3. a = -47/44, b = 9/44
4. a = 47/44, b = 9/44

Correct answer: a = -47/44, b = -9/44

Solution

Multiply by (1 + 3 sqrt5): denominator = 1 - (3 sqrt5)² = 1 - 45 = -44. Numerator = (2 + 3 sqrt5)(1 + 3 sqrt5) = 2 + 6 sqrt5 + 3 sqrt5 + 45 = 47 + 9 sqrt5. So expression = (47 + 9 sqrt5)/(-44) = -47/44 - (9/44) sqrt5. Hence a = -47/44, b = -9/44.

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