En résolvant l’équation quadratique, \( n = \frac{-7 \pm \sqrt{7^2 + 4 \cdot 3 \cdot 220}}{2 \cdot 3} = \frac{-7 \pm 37}{6} \).
["Solving the Quadratic Equation: A Complete Guide to ( n = \frac{-7 \pm \sqrt{7^2 + 4 \cdot 3 \cdot 220}}{2 \cdot 3} )", "Learning how to solve quadratic equations is a fundamental skill in algebra, and understanding the quadratic formula can simplify complex problems. In this article, we’ll break down the step-by-step process of solving the specific quadratic equation:\n[\nn = \frac{-7 \pm \sqrt{7^2 + 4 \cdot 3 \cdot 220}}{2 \cdot 3}\n]", "This equation appears in real-world modeling, physics, and optimization problems, making its resolution both practical and educational.", "---", "### Understanding the Quadratic Formula\nThe general form of a quadratic equation is:\n[\nax^2 + bx + c = 0\n]\nThe solutions for ( x ) are given by the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nIn our case, we are solving for ( n ) (not ( x )), and the equation matches the standard form with:\n- ( a = 3 )\n- ( b = -7 )\n- ( c = 220 )", "---", "### Step-by-Step Calculation", "Start with the discriminant:\n[\nD = b^2 + 4ac = (-7)^2 + 4 \cdot 3 \cdot 220\n]\nCompute each term:\n- ( (-7)^2 = 49 )\n- ( 4 \cdot 3 \cdot 220 = 2640 )\nSo,\n[\nD = 49 + 2640 = 2689\n]", "Now substitute into the quadratic formula:\n[\nn = \frac{-(-7) \pm \sqrt{2689}}{2 \cdot 3} = \frac{7 \pm \sqrt{2689}}{6}\n]\nWait — note: earlier the expression claims ( +4ac ), but standard formula uses ( b^2 - 4ac ). Let's verify:\nWe computed ( D = b^2 - 4ac ) correctly as:\n[\nD = (-7)^2 - 4(3)(220) = 49 - 2640 = -2591\n]\nWait — here lies a critical error! Since the original equation was written as:\n[\nn = \frac{-7 \pm \sqrt{7^2 + 4 \cdot 3 \cdot 220}}{2 \cdot 3}\n]\nthis assumes the discriminant is ( b^2 + 4ac ), which violates standard form.", "---", "### Correcting the Equation\nThe correct discriminant is:\n[\nD = b^2 - 4ac = (-7)^2 - 4 \cdot 3 \cdot 220 = 49 - 2640 = -2591\n]\nSince the discriminant is negative, this quadratic has no real solutions — only complex ones.", "Proceeding formally,\n[\n\sqrt{D} = \sqrt{-2591} = i\sqrt{2591}\n]\nThus, the two complex solutions are:\n[\nn = \frac{7 \pm i\sqrt{2591}}{6}\n]\nThis showing the importance of checking the discriminant before finalizing answers.", "---", "### Why This Matters in Real Problems\nEquations like this emerge in disciplines such as:\n- Engineering, where quadratic models predict optimal configurations\n- Physics, when analyzing motion under constant acceleration\n- Economics, when modeling profit functions with quadratic cost", "Even with complex results, understanding discriminants helps classify solutions and decide next steps.", "---", "### Final Answer:\n[\nn = \frac{7 \pm i\sqrt{2591}}{6}\n]\nThere are no real solutions due to a negative discriminant, but the full complex solution set is fully derivable using the quadratic formula with proper sign handling.", "---", "### Key Takeaways\n- Always simplify the discriminant carefully — ( b^2 - 4ac ), not addition.\n- The sign in the formula (( -b )) is crucial — remember ( b = -7 ), so ( -b = +7 ).\n- Complex numbers extend solution scope when discriminant is negative.\n- Real-world applications benefit from accurate algebraic reductions.", "Mastering these principles turns abstract math into powerful problem-solving tools.", "---", "Keywords: quadratic equation solution, solving quadratic, discriminant calculation, real and complex roots, quadratic formula tutorial, solving ( n = \frac{-7 \pm \sqrt{...}}{6} ), complex solutions in quadratics.", "Meta Title: How to Solve ( n = \frac{-7 \pm \sqrt{7^2 + 4 \cdot 3 \cdot 220}}{2 \cdot 3} ) — Step-by-Step Guide\nMeta Description: Learn to solve the quadratic equation ( n = \frac{-7 \pm \sqrt{7^2 + 4 \cdot 3 \cdot 220}}{2 \cdot 3} ) with accurate discriminant handling and real/complex outcome explanation."]
