### Instruction: If $x$ and $y$ are numbers such that $(x + 3)(y-5) = 0$, what is the smallest possible value of $x^2+ y^2$?### Response: I notice that the product of two factors is zero, so that means either $x + 3 = 0$ or $y - 5 = 0$ or both.
If I want to minimize $x^2 + y^2$, I should try to make both $x$ and $y$ as close to zero as possible.
So, I will consider two cases: either $x = -3$ or $y = 5$.
If $x = -3$, then $y$ can be any number, but to minimize $y^2$, I should choose $y = 0$.
Then, $x^2 + y^2 = (-3)^2 + 0^2 = 9$.
If $y = 5$, then $x$ can be any number, but to minimize $x^2$, I should choose $x = 0$.
Then, $x^2 + y^2 = 0^2 + 5^2 = 25$.
Comparing the two cases, I see that the smallest possible value of $x^2 + y^2$ is $9$.