Question

The geometric and arithmetic mean of the two numbers are 8 and 17 respectively.
1. Find the bigger number.
2. Find the smaller number.
3. Find the harmonic mean.

Answer 1

let the two numbers be x and y

(x+y)/2 = 17
x+y = 34 --> y = 34-x

√(xy) = 8
xy=64
x(34-x) = 64
x^2 - 34x = -64
x^2 - 34x + 289 = 289-64 ---> I completed the square
(x-17)^2 = 225
x-17 = ±15
x = 32 or x = 2

if x = 32, y = 2
if x = 2, y = 32 , (called symmetric solution)

The bigger is 32, the smaller is 2

If I recall correctly, the harmonic mean would be
2/(1/2 + 1/32)
= 2/( 17/32)
= 64/17