Question

webwork / f2025math265 / webwork 4 - topics 8 - 9 / 3
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webwork 4 - topics 8 - 9: problem 3
(1 point)
find two numbers differing by 30 whose product is as small as possible.
enter your two numbers as a comma separated list, e.g. 2, 3.
the two numbers are

Explanation:

Step1: Define variables

Let the two numbers be \(x\) and \(y\). Assume \(x>y\), so \(x - y=30\), then \(x = y + 30\). The product \(P=xy=y(y + 30)=y^{2}+30y\).

Step2: Find the minimum of the quadratic function

For a quadratic function \(f(y)=ay^{2}+by + c\) (\(a = 1\), \(b = 30\), \(c = 0\)), the vertex of the parabola \(y=-\frac{b}{2a}\).
Substitute \(a = 1\) and \(b = 30\) into \(y=-\frac{b}{2a}\), we get \(y=-\frac{30}{2\times1}=- 15\).

Step3: Find the other number

Since \(x=y + 30\), when \(y=-15\), \(x=-15 + 30=15\).

Answer:

\(-15,15\)