1. **State the problem:** We are given two integers $x$ and $y$ such that their difference is $x - y = -18$. We want to find: a) The minimum product of the two numbers. b) The two numbers that give this minimum product. 2. **Express the relationship:** From $x - y = -18$, we can rewrite $y$ in terms of $x$: $$y = x + 18$$ 3. **Write the product function:** The product $P$ of the two numbers is: $$P = x \times y = x(x + 18) = x^2 + 18x$$ 4. **Find the minimum product:** Since $P = x^2 + 18x$ is a quadratic function in $x$ with a positive leading coefficient, it opens upward and has a minimum at its vertex. The vertex of $ax^2 + bx + c$ is at $x = -\frac{b}{2a}$. Here, $a = 1$, $b = 18$, so: $$x = -\frac{18}{2 \times 1} = -9$$ 5. **Calculate the minimum product:** Substitute $x = -9$ back into $P$: $$P = (-9)^2 + 18(-9) = 81 - 162 = -81$$ 6. **Find the corresponding $y$:** $$y = x + 18 = -9 + 18 = 9$$ 7. **Answer:** a) The minimum product is $-81$. b) The two numbers are $-9$ and $9$. This means the product of the two integers is minimized when the numbers are $-9$ and $9$, giving a product of $-81$.