The sum of two numbers is 15 and their geometric mean is 20% lower than the…

Let the two numbers be x and y.

Given:

• x + y = 15, so the arithmetic mean (AM) = (x+y)/2 = 7.5.

• The geometric mean (GM) is 20% lower than AM, so GM = 0.8 × 7.5 = 6, i.e. √(xy) = 6 ⇒ xy = 36.

Now x and y are roots of the quadratic equation t^2 − (sum)t + (product) = 0, so:

1. t^2 − 15t + 36 = 0

2. Discriminant Δ = 15^2 − 4×36 = 225 − 144 = 81, so √Δ = 9.

3. Roots: t = (15 ± 9)/2 ⇒ t = 12 or t = 3.

Therefore the two numbers are 12 and 3.