Step-by-step explanation:
The arithmetic mean is (a + b) / 2.
The geometric mean is √(ab).
If the arithmetic mean is twice the geometric mean:
(a + b) / 2 = 2√(ab)
a + b = 4√(ab)
(a + b)² = 16ab
a² + 2ab + b² = 16ab
a² − 14ab + b² = 0
Complete the square:
a² − 14ab + 49b² + b² = 49b²
(a − 7b)² = 48b²
a − 7b = √48 b
a = (7 + √48) b
a/b = 7 + √48
a/b = 7 + 4√3
We can show this equals (2+√3)/(2−√3) by multiplying by (2−√3)/(2−√3).
a/b = (7 + 4√3) × (2 − √3) / (2 − √3)
a/b = (14 − 7√3 + 8√3 − 12) / (2 − √3)
a/b = (2 + √3) / (2 − √3)
