Final answer:
The test that determines if at least one root is within an interval is if the product of the start and end values of the interval is negative; this implies the function changes sign, indicating a root exists within the interval.
Step-by-step explanation:
The test used to determine whether at least one root is found within an interval is described by the statement: The product (multiplication) of the start and end values in the interval is negative when there is a root and positive when there is no root. This is based on the Intermediate Value Theorem, which in simple terms suggests that if you have a continuous function that takes on different signs at two points, there must be at least one root between them.
For example, if the function is positive at one end of the interval and negative at the other, then by continuity, the function must cross the x-axis at some point within the interval, indicating the presence of a root. Two-Dimensional (x-y) Graphing can visually confirm this when a curve crosses the x-axis, which signifies the location of a root. However, Quadratic equations constructed on physical data always have real roots, and these roots can be distinguished by their sign. It is important to remember that physically meaningful solutions usually correspond to positive roots.