Question

of all numbers whose difference is 16, find the two that have the minimum product.
what are the two numbers?
(use a comma to separate answers as needed.)

Explanation:

Step1: Define variables

Let the two numbers be \(x\) and \(y\). Given \(x - y=16\), so \(y=x - 16\).
The product \(P=xy=x(x - 16)=x^{2}-16x\).

Step2: Find the vertex of the quadratic function

For a quadratic function \(y = ax^{2}+bx + c\) (\(a = 1\), \(b=-16\), \(c = 0\) in \(P=x^{2}-16x\)), the \(x\) - coordinate of the vertex is given by \(x=-\frac{b}{2a}\).
Substitute \(a = 1\) and \(b=-16\) into \(x=-\frac{b}{2a}\), we get \(x=-\frac{-16}{2\times1}=8\).

Step3: Find the value of \(y\)

Since \(y=x - 16\), when \(x = 8\), \(y=8-16=-8\).

Answer:

\(8,-8\)