## Precalculus (6th Edition) Blitzer

The maximum product is $81$ and the pair is $\left( -9,-9 \right)$.
Let the two numbers be x and y. Then, $x\ +\ y\ =\ -18$ Now, \begin{align} & x+y=-18 \\ & y=-18-x \end{align} Calculate xy, \begin{align} & xy=x\left( -18-x \right) \\ & xy=-18x-{{x}^{2}} \\ \end{align} Which is a downwards opening parabola, which attains its maximum value at $\frac{-b}{2a}$ , where $b=-18$ and $a=-1$. \begin{align} & x=\frac{18}{-2} \\ & x=-9 \\ \end{align} Substituting the value of x in the equation (1) we get, \begin{align} & y\ =\ \ -\left( -9 \right)\ \ -\ 18 \\ & =\ \ 9\ \ -\ \ 18 \\ & =\ \ -9 \end{align} The maximum product is: \begin{align} & xy\ =\ \left( -9 \right)\ \left( -9 \right) \\ & =\ 81 \end{align} The maximum product is $81$ and the pair is $\left( -9,\ -9 \right)$.