Question

two numbers have a difference of 8. if the product of the two numbers if a minimum, then what are the two numbers?

Explanation:

Step1: Define variables

Let the two numbers be \(x\) and \(y\). Assume \(x - y=8\), so \(y=x - 8\).

Step2: Set up the product function

The product \(P=xy=x(x - 8)=x^{2}-8x\).

Step3: Find the vertex of the quadratic function

For a quadratic function \(y = ax^{2}+bx + c\) (\(a = 1\), \(b=-8\), \(c = 0\)), the \(x\) - coordinate of the vertex is \(x=-\frac{b}{2a}\).
Substitute \(a = 1\) and \(b=-8\) into \(x=-\frac{b}{2a}\), we get \(x=-\frac{-8}{2\times1}=4\).

Step4: Find the other number

Since \(y=x - 8\), when \(x = 4\), \(y=4-8=-4\).

Answer:

The two numbers are \(4\) and \(-4\).