t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}

["Solving Quadratic Equations: A Step-by-Step Guide to the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, essential for students and professionals alike. One of the most powerful tools for finding the roots of any quadratic equation of the form\n[ ax^2 + bx + c = 0 ]\nis the quadratic formula:\n[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nIn this article, we’ll break down how this formula works, apply it to an example, and explore its significance in mathematics.", "---", "### Understanding the Quadratic Formula", "The quadratic formula solves general quadratic equations using values from the equation coefficients:\n- ( a ): Coefficient of ( x^2 )\n- ( b ): Coefficient of ( x )\n- ( c ): Constant term", "The formula is:\n[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Note: The expression under the square root — ( b^2 - 4ac ) — is known as the discriminant, which determines the nature of the roots:\n- Positive discriminant → Two distinct real roots\n- Zero discriminant → One real repeated root\n- Negative discriminant → Two complex conjugate roots", "Let’s rewrite your example equation clearly:\nIf ( a = 3 ), ( b = -4 ), and ( c = 1 ), the equation becomes:\n[\n3t^2 - 4t + 1 = 0\n]", "---", "### Applying the Quadratic Formula", "Substitute ( a = 3 ), ( b = -4 ), and ( c = 1 ) into the formula:\n[\nt = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}\n]", "Simplify step-by-step:", "1. ( -b = -(-4) = 4 )\n2. The discriminant:\n[\n(-4)^2 - 4 \cdot 3 \cdot 1 = 16 - 12 = 4\n]\n3. Square root of discriminant:\n[\n\sqrt{4} = 2\n]\n4. Denominator:\n[\n2 \cdot 3 = 6\n]\n5. Final expression:\n[\nt = \frac{4 \pm 2}{6}\n]", "---", "### Finding the Roots", "Using the ( \pm ) to find two solutions:", "- First root (with ( + )):\n[\nt_1 = \frac{4 + 2}{6} = \frac{6}{6} = 1\n]", "- Second root (with ( - )):\n[\nt_2 = \frac{4 - 2}{6} = \frac{2}{6} = \frac{1}{3}\n]", "---", "### Summary of Results", "For the quadratic equation\n[\n3t^2 - 4t + 1 = 0,\n]\nthe solutions are:\n- ( t = 1 )\n- ( t = \frac{1}{3} )", "These values satisfy the original equation, confirming that the quadratic formula provides accurate results.", "---", "### Why the Quadratic Formula Matters", "Beyond algebra, this formula enables solving real-world problems involving parabolas—such as projectile motion, optics, and economics. It unifies the approach to solving any quadratic equation regardless of factoring ease.", "---", "### Practice: Try It Yourself", "1. Identify coefficients ( a, b, c ) in ( ax^2 + bx + c = 0 ).\n2. Plug into the quadratic formula cautiously — remember signs!\n3. Simplify step-by-step to avoid errors.", "---", "In conclusion, understanding the quadratic formula —\n[\nt = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\n—is crucial for mastering algebra. With practice, you’ll not only solve equations faster but also deepen your problem-solving intuition.", "For further exploration, check out more advanced quadratics, including completing the square and applications like quadratic optimization.", "---", "Keywords: quadratic formula, solve quadratic equation, solve t = -(-4)√(...) + 3t + 1 = 0, discriminant, algebra tutorial, polynomial roots, quadratic formula steps, real roots, complex roots, math practice.", "Meta Description: Learn how to solve quadratic equations using the quadratic formula, step-by-step with example t = -(-4)√(...) + 3t + 1 = 0, and master key math skills."]