Answer
The pair is $-8$ and $8$.
The minimum product is $-64$.
Work Step by Step
The given difference of the numbers in the pair is $16$.
Let's note by $x$ one of the numbers of the pair whose product is minimum.
The other number is $x+16$.
The product is
$\Rightarrow P(x)=x(x+16)$
Use the distributive property.
$\Rightarrow P(x)=x^2+16x$.
The standard form of a quadratic equation is $f(x)=ax^2+bx+c$.
Compare both equations to identify the constants $a=1$, $b=16$, $c=0$.
The value of $a$ is greater than zero, the function has the minimum value.
The value of $x$ for which the minimum is reached is $-\frac{b}{2a}$.
Substitute all values.
$\Rightarrow x=-\frac{16}{2(1)}$.
Simplify.
$\Rightarrow x=-8$.
The other number is $x+16=-8+16=8$.
Hence, the pair is $-8$ and $8$.
The minimum product is $8(-8)=-64$.
