The roots are \( t = 2 \) and \( t = 4 \). The expression is less than or equal to zero between the roots:
["Title: Understanding When the Quadratic Expression Is Less Than or Equal to Zero: Roots at ( t = 2 ) and ( t = 4 )", "When analyzing quadratic expressions, identifying the roots and understanding where the function is less than or equal to zero is essential for solving inequalities and interpreting graph behavior. In this article, we explore a quadratic expression with roots at ( t = 2 ) and ( t = 4 ), and explain why the expression is less than or equal to zero between these two points.", "---", "### What Are the Roots of the Expression?", "For the given quadratic expression, the roots occur at ( t = 2 ) and ( t = 4 ). These values mean that when ( t ) equals 2 or 4, the expression equals zero:", "[\nf(t) = 0 \quad \ ext{when} \quad t = 2 \quad \ ext{or} \quad t = 4\n]", "Since the roots are distinct and ordered, the number line is divided into three intervals:", "1. ( t < 2 )\n2. ( 2 < t < 4 )\n3. ( t > 4 )", "---", "### Where Is the Expression Less Than or Equal to Zero?", "The quadratic expression changes sign at its roots depending on its leading coefficient (whether the parabola opens upwards or downwards). For simplicity, assume the quadratic opens upward (positive leading coefficient), consistent with most standard upward-facing parabolas.", "Between the roots ( t = 2 ) and ( t = 4 ), the graph lies below or touches the x-axis—making the expression less than or equal to zero. Thus,", "[\nf(t) \leq 0 \quad \ ext{for} \quad 2 \leq t \leq 4\n]", "This interval represents the set of values where the quadratic is negative or zero.", "---", "### Visualizing the Behavior", "- To the left of ( t = 2 ): ( f(t) > 0 ) (positive values)\n- At ( t = 2 ) and ( t = 4 ): ( f(t) = 0 ) (roots)\n- Between ( t = 2 ) and ( t = 4 ): ( f(t) < 0 ) (negative values)\n- To the right of ( t = 4 ): ( f(t) > 0 ) again", "You can plot this parabola to see how it dips below zero between the two roots.", "---", "### Real-World Implications", "Understanding where a quadratic expression is less than or equal to zero has practical meaning in many fields:", "- Engineering: Modeling stress or displacement thresholds\n- Economics: Identifying profit or loss regions beyond break-even points\n- Physics: Analyzing motion where velocity or position constraints apply", "For our expression, any application between ( t = 2 ) and ( t = 4 ) requires attention to values where the outcome is non-positive.", "---", "### How to Confirm Using the Standard Form", "If the quadratic expression is written as:", "[\nf(t) = a(t - 2)(t - 4), \quad a > 0\n]", "Expanding gives ( f(t) = a(t^2 - 6t + 8) ), confirming a positive leading coefficient and a U-shaped graph with roots at 2 and 4. This confirms our earlier conclusion: ( f(t) \leq 0 ) between ( t = 2 ) and ( t = 4 ).", "---", "### Summary", "- The roots ( t = 2 ) and ( t = 4 ) divide the number line into intervals.\n- For ( 2 \leq t \leq 4 ), the expression is less than or equal to zero.\n- Below or at ( t = 2 ) and ( t = 4 ), the value is zero; in between, it's negative under upward-opening parabolas.\n- Recognizing this interval supports solving inequalities and real-world modeling.", "By mastering the behavior of quadratics between roots, you build a strong foundation for analyzing functions and applications across science, engineering, and mathematics.", "---", "Keywords: quadratic inequality, roots ( t = 2 ), roots ( t = 4 ), ( f(t) \leq 0 ), analyzing quadratic expressions, understanding parabolas, solving inequalities, real-world applications."]
