Question: A science communicator designs an exhibit where the number of interactive stations $ S $ at time $ t $ (in hours) satisfies $ S(t) = t^2 - 5t + 6 $. Find all times $ t $ when $ S(t) = 0 $.

["Title: When Do Interactive Science Stations Total Zero? Solving $ S(t) = t^2 - 5t + 6 $", "In science exhibits where public engagement is measured by interactive stations, understanding when these stations are fully inactive or "turned off" is crucial for maintenance, scheduling, and visitor experience. Consider the function modeling the number of active stations over time:\n[\nS(t) = t^2 - 5t + 6\n]\nBut what does it mean when $ S(t) = 0 $? This article explores the times $ t $—in hours—when no interactive stations are active, helping exhibit designers optimize layout and timing.", "### Setting the Equation: When Are There Zero Active Stations?\nTo find when the number of interactive stations drops to zero, solve:\n[\nt^2 - 5t + 6 = 0\n]", "This is a quadratic equation. Using the quadratic formula:\n[\nt = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2}\n]\n[\nt = \frac{5 \pm 1}{2}\n]\nSo the solutions are:\n[\nt = \frac{5 + 1}{2} = 3 \quad \ ext{and} \quad t = \frac{5 - 1}{2} = 2\n]", "### Interpretation: Insertion and Exit Times\nThese values—$ t = 2 $ and $ t = 3 $—signal key moments:\n- At $ t = 2 $ hours, the exhibit activates its first set of interactive stations (enters active state).\n- At $ t = 3 $ hours, those stations shut down or reset, leaving no interactive stations at that moment.", "The number of stations transitions from active to inactive precisely at these times.", "### Analyzing Behavior Around the Roots\nThe quadratic opens upwards ($ t^2 $ coefficient positive), so $ S(t) $ is zero at $ t = 2 $ and $ t = 3 $, negative between them, and positive outside. This means:\n- $ S(t) < 0 $ between $ t = 2 $ and $ t = 3 $ (theoretical, possibly indicating system glitches or transitional phases).\n- The exhibit has zero active stations exactly at $ t = 2 $ and $ t = 3 $, providing clear timing cues for operations.", "### Practical Use for Science Communicators\nMonitoring $ S(t) = 0 $ allows staff to:\n- Plan maintenance during silent hours.\n- Alert visitors when interactive experiences are paused.\n- Refine station schedules based on real-time engagement patterns.", "### Conclusion\nThe interactive science exhibit experiences zero active stations precisely at $ t = 2 $ hours and $ t = 3 $ hours. Understanding these moments improves exhibit reliability and enhances visitor engagement. Science communicators can leverage this data to design smarter, responsive science experiences.", "Keywords: $ S(t) = t^2 - 5t + 6 $, interactive science exhibit, timing of station activation, solving quadratic equations, science communication, exhibit maintenance, bridge moment in engagement data"]