["# Solving the Inequality: -3t² - 4t + 12 > 0", "Understanding quadratic inequalities is a fundamental part of algebra, especially when dealing with real-world applications like physics, economics, and optimization problems. In this article, we explore the inequality -3t² - 4t + 12 > 0, break down how to solve it step-by-step, and interpret the solution in a meaningful context.", "---", "## What Does the Inequality Mean?", "We are given:", "$$
\n-3t^2 - 4t + 12 > 0
\n$$", "This represents a quadratic function in terms of ( t ), opening downward because the coefficient of ( t^2 ) is negative (( -3 )). We want to find all values of ( t ) where this quadratic value is greater than zero — that is, the section(s) of the parabola that lie above the ( t )-axis.", "---", "## Step 1: Rewrite the Inequality", "To solve:", "$$
\n-3t^2 - 4t + 12 > 0
\n$$", "We rewrite it in standard form by moving everything to one side — but since the inequality is already set to greater than zero, we proceed by solving the equality first:", "$$
\n-3t^2 - 4t + 12 = 0
\n$$", "---", "## Step 2: Solve the Corresponding Quadratic Equation", "We solve:", "$$
\n-3t^2 - 4t + 12 = 0
\n$$", "To simplify, multiply both sides by ( -1 ) (remember to reverse the inequality sign if we were managing inequalities, but here we focus on roots):", "$$
\n3t^2 + 4t - 12 = 0
\n$$", "Use the quadratic formula:", "$$
\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \ ext{where } a = 3, , b = 4, , c = -12
\n$$", "Calculate the discriminant:", "$$
\n\Delta = 4^2 - 4(3)(-12) = 16 + 144 = 160
\n$$", "Now compute the roots:", "$$
\nt = \frac{-4 \pm \sqrt{160}}{2 \cdot 3} = \frac{-4 \pm \sqrt{16 \cdot 10}}{6} = \frac{-4 \pm 4\sqrt{10}}{6} = \frac{-2 \pm 2\sqrt{10}}{3}
\n$$", "So the two real roots are:", "$$
\nt_1 = \frac{-2 - 2\sqrt{10}}{3} \approx \frac{-2 - 6.32}{3} \approx -2.77
\n$$
\n$$
\nt_2 = \frac{-2 + 2\sqrt{10}}{3} \approx \frac{-2 + 6.32}{3} \approx 1.44
\n$$", "---", "## Step 3: Analyze the Parabola and Determine Where the Inequality Holds", "Because the parabola opens downward (since coefficient of ( t^2 ) is negative), the graph is a “frown.” The quadratic expression is greater than zero between the two roots — that is, the expression is positive between the intercepts.", "Thus, the solution to:", "$$
\n-3t^2 - 4t + 12 > 0
\n$$", "is the open interval:", "$$
\n\boxed{ \left( \frac{-2 - 2\sqrt{10}}{3},\ \frac{-2 + 2\sqrt{10}}{3} \right) }
\n$$", "Or approximately:", "$$
\n\boxed{ (-2.77,\ 1.44) }
\n$$", "---", "## Step 4: Interpreting the Solution", "Graphically, this means that the quadratic function crosses the ( t )-axis at approximately ( t = -2.77 ) and ( t = 1.44 ), dips below the axis between those points, and becomes positive again only in the interval between them. Since the parabola opens downward, the function is positive strictly between the roots.", "---", "## Why This Matters", "Understanding where a quadratic expression is positive helps in numerous real-world scenarios:", "- Profit Maximization: When modeling profit as a quadratic function of production level ( t ), the inequality identifies production ranges that yield positive (or profitable) returns.
\n- Physics: Computation of motion timelines, projectile paths, or energy thresholds often involve quadratic inequalities.
\n- Economics & Business: Analyzing break-even points and intervals of profitability.", "---", "## Conclusion", "Solving the inequality ( -3t^2 - 4t + 12 > 0 ) involved finding the roots of the associated quadratic equation and leveraging the shape of the parabola to determine the interval where the expression is positive. The solution set is an open interval between the two real roots, reflecting the downward-opening nature of the function.", "Mastering quadratic inequalities empowers you to model complex real-life thresholds and optimize decisions — a key skill in STEM and applied mathematics.", "---", "See also:
\n- How to solve quadratic inequalities
\n- Graphing quadratic expressions
\n- Real-world applications of parabolic functions", "---", "Keywords: quadratic inequality, solving -3t² - 4t + 12 > 0, finding roots, quadratic function graph, real-world applications, algebra tutorial, tiez645t² inequality solution."]
