Question

of all numbers whose difference is 18, find the two that have the minimum product. what are the two numbers? (use a comma to separate answers as needed.)

Explanation:

Step1: Set up variables

Let the two numbers be \(x\) and \(y\). Assume \(x>y\), so \(x - y=18\), then \(x=y + 18\). The product \(P=xy=y(y + 18)=y^{2}+18y\).

Step2: Find the minimum of the quadratic function

For a quadratic function \(P(y)=ay^{2}+by + c\) (\(a = 1\), \(b = 18\), \(c = 0\)), the vertex of the parabola \(y=-\frac{b}{2a}\).
Substitute \(a = 1\) and \(b = 18\) into the formula \(y=-\frac{18}{2\times1}=-9\).

Step3: Find the other number

Since \(x=y + 18\), when \(y=-9\), \(x=-9 + 18=9\).

Answer:

\(9,-9\)