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Let's call the two numbers x and y.

The arithmetic mean of x and y is defined as the sum of the two numbers divided by 2:

(x + y)/2 = 40

Multiplying both sides by 2, we get:

x + y = 80

The geometric mean of x and y is defined as the square root of their product:

sqrt(x*y) = 32

Squaring both sides, we get:

x*y = 32^2 = 1024

We now have two equations:

x + y = 80

x*y = 1024

We can use these equations to solve for x and y.

One way to do this is to use substitution. Rearrange the first equation to solve for one of the variables in terms of the other:

x + y = 80

y = 80 - x

Substitute this expression for y in the second equation:

x*y = 1024

x*(80 - x) = 1024

Expanding the left side, we get:

80x - x^2 = 1024

Rearranging terms and setting equal to zero, we get a quadratic equation:

x^2 - 80x + 1024 = 0

We can solve for x using the quadratic formula:

x = [80 +/- sqrt(80^2 - 411024)] / 2

x = [80 +/- sqrt(384)] / 2

x = [80 +/- 16sqrt(6)] / 2

x = 40 +/- 8sqrt(6)

We get two solutions for x:

x = 40 + 8sqrt(6) ≈ 66.66

or

x = 40 - 8sqrt(6) ≈ 13.34

We can use either of these values to solve for y using the equation y = 80 - x:

If x = 40 + 8sqrt(6):

y = 80 - x = 80 - (40 + 8sqrt(6)) = 40 - 8sqrt(6) ≈ 13.34

If x = 40 - 8sqrt(6):

y = 80 - x = 80 - (40 - 8sqrt(6)) = 40 + 8sqrt(6) ≈ 66.66

Therefore, the two numbers are approximately 13.34 and 66.66.

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