AM and GM

For positive numbers a and b, the arithmetic mean is A = (a + b)/2 and the geometric mean is G = √(ab).

A − G = (a + b − 2√(ab))/2 = (√a − √b)²/2, which is never negative. So A ≥ G for all positive a and b.

Equality A = G holds only when √a = √b, that is when a = b. For 2 and 8, A = 5 and G = 4.

The two means fix the numbers. If A = 10 and G = 8 then a + b = 20 and ab = 64, so (a − b)² = 400 − 256 = 144, a − b = ±12, and the numbers are 4 and 16.

Another pair: A = 13 and G = 12 give a + b = 26, ab = 144, (a − b)² = 676 − 576 = 100, so the numbers are 8 and 18.

Numbers with A and G known are the roots of x² − 2Ax + G² = 0. For A = 5 and G = 4 that is x² − 10x + 16 = 0, with roots 2 and 8.

A ≥ G gives quick bounds: for positive x, x + 1/x ≥ 2; two positive numbers with product 36 have sum at least 12; a rectangle with perimeter 40 has area at most 10 × 10 = 100.