This is a quadratic equation in the form \( at^2 + bt + c = 0 \), where \( a = 3 \), \( b = -2 \), and \( c = -15 \). We use the quadratic formula:

["Solving the Quadratic Equation ( 3t^2 - 2t - 15 = 0 ) Using the Quadratic Formula", "Quadratic equations are foundational in algebra and appear in diverse fields such as physics, engineering, economics, and computer science. Understanding how to solve them not only helps in academic settings but also strengthens problem-solving skills. This article focuses on solving the specific quadratic equation in standard form:", "[\n3t^2 - 2t - 15 = 0\n]", "where ( a = 3 ), ( b = -2 ), and ( c = -15 ). We’ll explore the method of the quadratic formula, provide a detailed step-by-step solution, and help you grasp how to approach similar problems.", "---", "### What Is a Quadratic Equation?", "A quadratic equation is any polynomial equation of degree 2, meaning the highest exponent of the variable ( t ) is 2. The general form is:", "[\nat^2 + bt + c = 0\n]", "with ( a <br/>\neq 0 ). The solution depends on the discriminant ( D = b^2 - 4ac ), which determines the number and nature of the roots:", "- If ( D > 0 ): Two distinct real solutions\n- If ( D = 0 ): One real (repeated) solution\n- If ( D < 0 ): Two complex (imaginary) solutions", "---", "### Applying the Quadratic Formula", "To solve any quadratic equation, the quadratic formula provides the exact roots:", "[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging in ( a = 3 ), ( b = -2 ), and ( c = -15 ), we proceed step by step.", "---", "### Step-by-Step Solution", "Step 1: Write down the coefficients\n[\na = 3, \quad b = -2, \quad c = -15\n]", "Step 2: Calculate the discriminant ( D )\n[\nD = b^2 - 4ac = (-2)^2 - 4(3)(-15) = 4 + 180 = 184\n]\nSince ( D = 184 > 0 ), there are two distinct real solutions.", "Step 3: Plug values into the quadratic formula\n[\nt = \frac{-(-2) \pm \sqrt{184}}{2 \cdot 3} = \frac{2 \pm \sqrt{184}}{6}\n]", "Step 4: Simplify ( \sqrt{184} )\nFactor 184 to simplify the square root:\n[\n184 = 4 \ imes 46 \Rightarrow \sqrt{184} = \sqrt{4 \ imes 46} = 2\sqrt{46}\n]", "So the solution becomes:\n[\nt = \frac{2 \pm 2\sqrt{46}}{6}\n]", "Step 5: Reduce the fraction\nFactor out 2 in numerator:\n[\nt = \frac{2(1 \pm \sqrt{46})}{6} = \frac{1 \pm \sqrt{46}}{3}\n]", "---", "### Final Answer", "Thus, the solutions to the quadratic equation ( 3t^2 - 2t - 15 = 0 ) are:", "[\nt = \frac{1 + \sqrt{46}}{3} \quad \ ext{and} \quad t = \frac{1 - \sqrt{46}}{3}\n]", "These are two distinct real numbers. Numerically, they approximate to about ( t \approx 2.76 ) and ( t \approx -1.87 ), respectively.", "---", "### Why Learn This Method?", "Mastering the quadratic formula equips you with a powerful tool applicable in countless real-world scenarios:", "- Physics: Modeling projectile motion and quadratic relationships in forces.\n- Engineering: Calculating optimal designs involving parabolic trajectories or stress analysis.\n- Finance: Solving problems involving profit functions and break-even analysis.", "---", "### Comparing with the Simplified Form", "The original equation ( 3t^2 - 2t - 15 = 0 ) differs slightly in coefficient signs from a more typical format. However, understanding how discriminant and term signs affect the formula remains crucial. When coefficients are adjusted, always recalculate ( D ), ( a ), and ( b ) carefully to ensure accurate results.", "---", "### Conclusion", "Solving quadratic equations using the quadratic formula is a reliable and universal method. By identifying coefficients ( a ), ( b ), and ( c ) clearly, computing the discriminant, and simplifying step-by-step, you can solve equations like ( 3t^2 - 2t - 15 = 0 ) efficiently. Whether in school or in professional fields, this skill builds a strong foundation in algebra and analytical reasoning.", "---", "### More Practice Tips", "- Always double-check calculations, especially with fractions and square roots.\n- Recognize patterns: Sometimes factoring or completing the square may simplify problems, but the quadratic formula is always reliable.\n- Practice with varied values to build confidence in applying the method systematically.", "---", "Keyword-rich takeaway: quadratic equation solution using quadratic formula with ( a=3, b=-2, c=-15 ), direct calculation, real roots via discriminant, simplified radical form — essential step-by-step algebra skill."]