The real numbers x and y satisfy x (x+y) = 27 and (x+y) = 54. What is the value of (x+y)^2?
We have $(x+y)^2 = (x(x+y))^2 = 27^2 = \boxed{729}$.
To find the value of (x+y)^2, we need to use the equation x (x+y) = 27.
From the given information, we also have that the value of (x+y) is 54.
We can substitute this value into the given equation to solve for x:
x (x+y) = 27
x * 54 = 27
Dividing both sides of the equation by 54:
x = 27/54
Simplifying, we get:
x = 1/2
Now that we know the value of x, we can substitute it back into the equation (x+y) = 54 to solve for y:
1/2 + y = 54
Subtracting 1/2 from both sides of the equation:
y = 54 - 1/2
Simplifying, we get:
y = 53 1/2
Now we have the values of x and y. To find the value of (x+y)^2, we simply substitute these values into the equation:
(x+y)^2 = (1/2 + 53 1/2)^2
Simplifying, we get:
(1/2 + 53 1/2)^2 = (54)^2
Therefore, the value of (x+y)^2 is 54^2, which is equal to 2916.