Exercise 33 Page 234

Let $x$ and $y$ be the desired real numbers. Using the given information, we can write two equations, one for the sum of the numbers and one for the product of the numbers. We can solve this system of equations using the Substitution Method. First, let's isolate $y$ in the first equation and then substitute it into the second equation. Our next step is to solve the obtained quadratic equation. We can do this by graphing. Let's begin by finding the axis of symmetry of the parabola using the formula $x=\text{-}\frac{b}{2a}.$ For our equation, we have that $a$ $=$ $\text{-}1$ and $b$ $=$ $\text{-}15.$ Next, we will make a table of values using $x$ values around the axis of symmetry $x=\text{-}7.5.$

$x$	$\text{-}{x}^2-15x+54$	$y$
$\text{-}18$	$\text{-}(\text{-}18{)}^2-15(\text{-}18)+54$	$0$
$\text{-}16$	$\text{-}(\text{-}16{)}^2-15(\text{-}16)+54$	$38$
$\text{-}7.5$	$\text{-}(\text{-}7.5{)}^2-15(\text{-}7.5)+54$	$110.25$
$1$	$\text{-}(1{)}^2-15(1)+54$	$38$
$3$	$\text{-}(3{)}^2-15(3)+54$	$0$

Now, we will plot and connect the obtained points.

The roots of the equation are $\text{-}18$ and $3.$ Let's investigate if they satisfy the original problem. Since the numbers satisfy both requirements stated in the exercise we have found the numbers we are looking for, and they are $\text{-}18$ and $3.$