Final answer:
The minimum value of the expression x^2 + y^2 - 6x + 4y + 18 is 5.
Step-by-step explanation:
The given expression is a quadratic equation in terms of x and y.
To find the minimum value of the expression, we can complete the square for both x and y. Let's start by completing the square for x:
x^2 - 6x = (x^2 - 6x + 9) - 9 = (x - 3)^2 - 9
Now, let's complete the square for y:
y^2 + 4y = (y^2 +4y + 4) - 4 = (y + 2)^2 - 4
Substituting back into the original expression, we get:
(x - 3)^2 + (y + 2)^2 - 9 - 4 + 18 = (x - 3)^2 + (y + 2)^2 + 5
The expression (x - 3)^2 + (y + 2)^2 + 5 represents the equation of a circle with center (3, -2) and radius sqrt(5).
Since the minimum value of a sum of squares is 0, the minimum value of the expression is 5.