Answer
The pair is $-12$ and $12$.
The minimum product is $-144$.
Work Step by Step
The given difference of the numbers in the pair is $24$.
Let's note by $x$ one of the numbers of the pair whose product is minimum.
The other number is $x+24$.
The product is
$\Rightarrow P(x)=x(x+24)$
Use the distributive property.
$\Rightarrow P(x)=x^2+24x$.
The standard form of a quadratic equation is $f(x)=ax^2+bx+c$.
Compare both equations to identify the constants $a=1$, $b=24$, $c=0$.
The value of $a$ is greater than zero, so the function has the minimum value.
The value of $x$ for which the minimum is reached is $-\frac{b}{2a}$.
Substitute all values.
$\Rightarrow x=-\frac{24}{2(1)}$.
Simplify.
$\Rightarrow x=-12$.
The other number is $x+24=-12+24=12$.
Hence, the pair is $-12$ and $12$.
The minimum product is $12(-12)=-144$.
