Two numbers differ by 6 and have a product of -9.

x - y = 6

x * y = -9

x = 3
y = -3

To solve this problem, let's assume the two numbers are x and y.

According to the problem, the two numbers differ by 6. Mathematically, we can express this as:

x - y = 6     (1)

Furthermore, the product of the two numbers is -9. We can express this relationship as:

x * y = -9     (2)

Now we have a system of two equations with two variables. We can use either the substitution or elimination method to solve them.

Let's solve the first equation (1) for x and substitute it into equation (2):

x = y + 6

Substituting x in equation (2):

(y + 6) * y = -9

Expanding the equation:

y^2 + 6y = -9

Rearranging terms:

y^2 + 6y + 9 = 0

This is a quadratic equation. We can solve it by factoring or using the quadratic formula. In this case, the equation factors as:

(y + 3)^2 = 0

Taking the square root of both sides:

y + 3 = 0

Solving for y:

y = -3

Now that we have the value of y, we can substitute it back into equation (1) to find x:

x - (-3) = 6

x + 3 = 6

x = 6 - 3

x = 3

So, the two numbers that differ by 6 and have a product of -9 are 3 and -3.