If (a + b√3)^2 = 52 + 30√3, where a,b natural, find a+b:
Question:
If (a + b√3)^2 = 52 + 30√3, where a,b natural, find a+b:
If (a + b√3)^2 = 52 + 30√3, where a,b natural, find a+b:
Options
Answer: 8
Explanation:
(a + b√3)² = 52 + 30√3 Expand the left side: a² + 2ab√3 + 3b² = 52 + 30√3 Match coefficients: 2ab = 30 → ab = 15 a² + 3b² = 52 Now try natural number pairs whose product is 15: a = 1, b = 15 → a² + 3b² = 1 + 3(225) = 676 (not 52) a = 3, b = 5 → 9 + 75 = 84 (not 52) a = 5, b = 3 → 25 + 27 = 52 ✅ So, a = 5 and b = 3. Then, a + b = 8. Answer: 8
Explanation:
(a + b√3)² = 52 + 30√3 Expand the left side: a² + 2ab√3 + 3b² = 52 + 30√3 Match coefficients: 2ab = 30 → ab = 15 a² + 3b² = 52 Now try natural number pairs whose product is 15: a = 1, b = 15 → a² + 3b² = 1 + 3(225) = 676 (not 52) a = 3, b = 5 → 9 + 75 = 84 (not 52) a = 5, b = 3 → 25 + 27 = 52 ✅ So, a = 5 and b = 3. Then, a + b = 8. Answer: 8
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