# What is the minimum possible product of two numbers that differ by 8 ?

Nov 6, 2016

The minimum product is $- 16$ and the two numbers are $- 4$ and $4$

#### Explanation:

Let the two numbers be $x$ and $x + 8$.

Then their product is:

$f \left(x\right) = x \left(x + 8\right)$

$\textcolor{w h i t e}{f \left(x\right)} = {x}^{2} + 8 x$

$\textcolor{w h i t e}{f \left(x\right)} = {x}^{2} + 8 x + 16 - 16$

$\textcolor{w h i t e}{f \left(x\right)} = {\left(x + 4\right)}^{2} - 16$

For any Real value of $x$ we will have ${\left(x + 4\right)}^{2} \ge 0$

Hence $f \left(x\right)$ attains its minimum value $- 16$ when ${\left(x + 4\right)}^{2} = 0$

That is when $x = - 4$

So the minimum product is $- 16$ and the two numbers are $- 4$ and $4$