Durch 4 teilen: \(x^2 + 115x - 780 = 0\)

["# Solving the Quadratic Equation ( x^2 + 115x - 780 = 0 ): A Step-by-Step Guide", "When faced with a quadratic equation like ( x^2 + 115x - 780 = 0 ), solving it systematically makes all the difference—especially for students, math enthusiasts, and educators. This article breaks down the equation into four clear, actionable steps using standard algebraic methods, offering practical tips for quick mastery. Whether you're preparing for exams or building foundational algebra skills, mastering this process will improve your problem-solving speed and accuracy.", "---", "## 1. Understand the Standard Form of the Equation", "Before diving into calculations, recognize the standard form of a quadratic equation:\n[ ax^2 + bx + c = 0 ]\nIn our example:\n- ( a = 1 ) (coefficient of ( x^2 ))\n- ( b = 115 )\n- ( c = -780 )", "This structure ensures consistency when applying solving techniques like factoring, completing the square, or the quadratic formula.", "---", "## 2. Apply the Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )", "With the equation in standard form, use the quadratic formula directly. Start by computing the discriminant ( D = b^2 - 4ac ):\n[\nD = (115)^2 - 4(1)(-780) = 13225 + 3120 = 16345\n]\nSince ( D > 0 ), there are two real solutions.\nNow, plug values into the formula:\n[\nx = \frac{-115 \pm \sqrt{16345}}{2}\n]\nWhile ( \sqrt{16345} ) is not a perfect square, approximate or compute it if needed (approximately ( 127.82 )).\nThus:\n[\nx \approx \frac{-115 + 127.82}{2} = \frac{12.82}{2} = 6.41\n]\n[\nx \approx \frac{-115 - 127.82}{2} = \frac{-242.82}{2} = -121.41\n]", "---", "## 3. Simplify and Verify Solutions", "Although decimals offer quick estimates, exact solutions are preferred. Since ( \sqrt{16345} ) simplifies only partially (as a product of primes), leaving the answer in radical form maintains precision:\n[\nx = \frac{-115 \pm \sqrt{16345}}{2}\n]\nTo verify, substitute each root back:\n- For ( x = \frac{-115 + \sqrt{16345}}{2} ):\n Plug into the original equation and confirm equality holds within rounding.\n- For ( x = \frac{-115 - \sqrt{16345}}{2} ):\n Same validation confirms correctness.", "---", "## 4. Practice Tips for Faster Mastery", "- Factor first when possible: Though large ( b ) and ( c ) suggest factoring might be tricky, check for integer factor pairs of ( -780 ) adding to ( 115 ).\n- Use a calculator early: To verify roots and discriminants quickly improves confidence.\n- Learn factoring tricks: For equations with large linear terms, use AC method or grouping.\n- Understand graph interpretation: Plotting the function ( y = x^2 + 115x - 780 ) helps visualize roots as x-intercepts.", "---", "## Summary", "Solving ( x^2 + 115x - 780 = 0 ) follows a structured four-step process: identify coefficients, calculate the discriminant, apply the quadratic formula, and verify results. While large numbers complicate exact solutions, mastering this method strengthens problem-solving fundamentals. Use this guide not only to solve the equation but to build a robust framework for tackling any quadratic equation efficiently.", "Mastering quadratic equations empowers you to excel in algebra, calculus, and applied fields—making consistent practice essential!"]