Given:
- Mean of the eight numbers = 10
- Variance of the eight numbers = 25

Formula:
- Mean (μ) = Sum of all numbers / Number of numbers
- Variance (σ²) = (Sum of squares of differences from the mean) / Number of numbers

Step 1: Calculate the sum of the eight numbers
Given numbers: 3, 7, 9, 12, 13, 20, x, y

Since we know the mean is 10, we can write the equation:
(3 + 7 + 9 + 12 + 13 + 20 + x + y) / 8 = 10

Multiplying both sides by 8, we get:
3 + 7 + 9 + 12 + 13 + 20 + x + y = 80

Simplifying the equation, we have:
x + y = 16 ----(1)

Step 2: Calculate the sum of squares of differences from the mean
We need to calculate the sum of squares of differences from the mean for all the eight numbers.

Given numbers: 3, 7, 9, 12, 13, 20, x, y

The sum of squares of differences from the mean can be written as:
(3 - 10)² + (7 - 10)² + (9 - 10)² + (12 - 10)² + (13 - 10)² + (20 - 10)² + (x - 10)² + (y - 10)² = 8 * 25

Simplifying the equation, we get:
(49 + 1 + 1 + 4 + 9 + 100 + (x - 10)² + (y - 10)²) = 200

Simplifying further, we have:
(x - 10)² + (y - 10)² = 26 ----(2)

Step 3: Solve the equations
We have two equations:
x + y = 16 ----(1)
(x - 10)² + (y - 10)² = 26 ----(2)

From equation (1), we can express y in terms of x:
y = 16 - x

Substituting this value in equation (2), we have:
(x - 10)² + (16 - x - 10)² = 26

Simplifying the equation, we get:
x² - 20x + 100 + 36 - 32x + x² = 26

Combining like terms, we obtain:
2x² - 52x + 110 = 0

Dividing the equation by 2, we have:
x² - 26x + 55 = 0

This equation can be factored as:
(x - 5)(x - 11) = 0

So, the possible values of x are 5 and 11.

Step 4: Calculate xy
For x = 5, y = 16 - x = 16 - 5 = 11
For x = 11, y = 16 - x =