Solution: Solve $ t^2 - 5t + 6 = 0 $. Factor: $ (t - 2)(t - 3) = 0 $. The solutions are $ t = 2 $ and $ t = 3 $. Thus, the times are $ oxed{2} $ and $ oxed{3} $.
["How to Solve the Quadratic Equation $ t^2 - 5t + 6 = 0 $ – Quick and Accurate Roots", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, and anyone working with mathematical models. One classic example is solving $ t^2 - 5t + 6 = 0 $. In this article, we’ll guide you through factoring, identifying solutions, and understanding the meaningful results—including the key times $ t = 2 $ and $ t = 3 $.", "## Why Solve $ t^2 - 5t + 6 = 0 $?", "Quadratic equations appear in many real-world applications, from physics and engineering to economics and computer science. Finding exact solutions allows precise predictions and informed decision-making. For this equation, the quadratic formula works, but factoring offers a faster, intuitive method.", "## Factoring $ t^2 - 5t + 6 $", "The expression $ t^2 - 5t + 6 $ can be factored into two binomials because it follows the form $ t^2 - (p + q)t + pq $, where $ p $ and $ q $ are constants. We look for two numbers that:", "- Multiply to $ +6 $ (the constant term), and\n- Add up to $ -5 $ (the coefficient of $ t $).", "The pair $ -2 $ and $ -3 $ satisfies both:\n$ (-2) \ imes (-3) = 6 $ and $ (-2) + (-3) = -5 $.", "Thus, we factor:\n$$\nt^2 - 5t + 6 = (t - 2)(t - 3)\n$$", "Setting the factored form equal to zero gives:\n$$\n(t - 2)(t - 3) = 0\n$$", "## Finding the Solutions", "By the zero-product property, if a product is zero, at least one factor must be zero. So:", "- $ t - 2 = 0 \Rightarrow t = 2 $\n- $ t - 3 = 0 \Rightarrow t = 3 $", "The two solutions are $ t = 2 $ and $ t = 3 $.", "## The Important Times: Boxed Results", "Therefore, the key times identified from this equation are:", "- Boxed answer: $ \boxed{2} $\n- Boxed answer: $ \boxed{3} $", "These values represent critical points—often interpreted as moment in time, values meeting a condition, or thresholds in modeling scenarios.", "## Summary", "To solve $ t^2 - 5t + 6 = 0 $, factor the quadratic into $ (t - 2)(t - 3) = 0 $. The solutions $ t = 2 $ and $ t = 3 $ are the points where the expression equals zero. Understanding and solving such equations builds a strong foundation in algebra with clear, practical applications.", "---", "Transform your math skills today—master factoring and unlock powerful problem-solving tools!"]
