t = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-15)}}{2 \cdot 3}

["Solving Quadratic Equations: Understanding the Quadratic Formula with a Detailed Example", "When tackling quadratic equations in algebra, the standard quadratic formula stands as a powerful tool to find the roots of any equation of the form ( at^2 + bt + c = 0 ). A common form encountered in mathematics involves expressions like:", "[\nt = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-15)}}{2 \cdot 3}\n]", "But what does this expression really mean? Let’s break it down step by step, explore its connection to the quadratic formula, and highlight how mastering this equation empowers problem-solving in science and engineering.", "---", "### Understanding the Quadratic Formula", "The general quadratic equation is written as:", "[\nat^2 + bt + c = 0\n]", "The quadratic formula provides the values of ( t ) that satisfy the equation:", "[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Notice how the expression inside the square root—called the discriminant (( b^2 - 4ac ))—determines the nature of the roots: real and distinct (positive discriminant), real and repeated (zero discriminant), or complex (negative discriminant).", "---", "### Analyzing the Given Expression", "The specific quadratic equation embedded in the expression is:", "[\n3t^2 + (-2)t + (-15) = 0\n]", "From this, we identify the coefficients:", "- ( a = 3 )\n- ( b = -2 )\n- ( c = -15 )", "Plugging these into the quadratic formula:", "[\nt = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-15)}}{2 \cdot 3}\n]", "Simplify step by step:", "- ( -(-2) = 2 )\n- ( (-2)^2 = 4 )\n- ( -4 \cdot 3 \cdot (-15) = 180 )\n- So the discriminant is ( 4 + 180 = 184 )", "Thus, the full expression becomes:", "[\nt = \frac{2 \pm \sqrt{184}}{6}\n]", "---", "### Simplifying the Square Root", "While ( \sqrt{184} ) is not a perfect square, it can be simplified:", "[\n\sqrt{184} = \sqrt{4 \cdot 46} = 2\sqrt{46}\n]", "So, the simplified solution is:", "[\nt = \frac{2 \pm 2\sqrt{46}}{6} = \frac{1 \pm \sqrt{46}}{3}\n]", "---", "### Why This Formula Matters", "Solving quadratic equations is essential across fields including physics, engineering, economics, and computer science. Applications include:", "- Projectile motion: determining peak height and landing time\n- Financial models: solving for interest rates or break-even points\n- Geometry: computing dimensions or distances", "Mastering the quadratic formula equips learners to tackle complex real-world problems used in design, optimization, and data analysis.", "---", "### Final Thoughts", "The formula:", "[\nt = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-15)}}{2 \cdot 3}\n]", "represents much more than symbolic manipulation — it’s a gateway to understanding how quadratic relationships model dynamic systems. Whether you're solving for ( t ) in a textbook or applying quadratic models in data science, knowing how to interpret and compute this expression gives you a solid mathematical foundation.", "Frequently, students struggle with sign errors and simplification, so always double-check signs of ( b ) and ( c ), and simplify radicals carefully. Practice with different coefficients builds fluency and confidence in solving quadratics.", "---", "Key Terms:\nQuadratic equation, quadratic formula, discriminant, algebraic simplification, applications of quadratics, solving ( t^2 ) equations.", "Automated SEO Keywords:\nquadratic formula derivation, solve quadratic equation, computational algebra, discriminant meaning, simplify square roots, quadratic formula example, algebraic solutions, quadratic roots calculation, projectile motion math, engineering applications quadratic"]