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.