Answer
the two numbers are $30$ and $30$
Work Step by Step
Let $x$ be the first number. Since the sum of the two numbers is $60,$ then the second number is $60-x$.
The product of the two numbers, $y,$ is
\begin{align*}
y&=x(60-x)
.\end{align*}
In the form $y=a(x-h)^2+k,$ the equation above is equivalent to
\begin{align*}
y&=60x-x^2
\\
y&=-x^2+60x
\\
y&=-(x^2-60x)
\\\\
y&=-\left(x^2-60x+\left(\dfrac{-60}{2}\right)^2\right)-(-1)\left(\dfrac{-60}{2}\right)^2
\\\\
y&=-\left(x^2-60x+900\right)+900
\\
y&=-\left(x-30\right)^2+900
.\end{align*}
Since the vertex of $y=a(x-h)^2+k$ is given by $(h,k)$, then the vertex of the equation above is $(30,900)$.
The maximum value occurs at the vertex $(h,k)$ where the maximum is $k$ when $x=h$. Thus the maximum product, $y,$ is $900$ which occurs when $x=30.$
Hence, the first number, $x,$ is $30$ and the second number ,$60-x$, is $30$.
