باستخدام القانون العام \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) حيث \( a = 1, b = 1, c = -420 \):

["Using the Quadratic Formula: Solving ( x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-420)}}{2(1)} ) Relaxed", "If you’re facing a quadratic equation and wondering how to find its solutions quickly and accurately, the universal formula you’ll turn to is the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "In real-world math and science applications, quadratic equations often take the form:", "[\nax^2 + bx + c = 0\n]\nor, when ( a = 1 ), simplify to:\n[\nx^2 + bx + c = 0\n]", "### Simplify Using the Given Values", "Let’s apply the quadratic formula step-by-step using the specific case:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "with ( a = 1 ), ( b = 1 ), and ( c = -420 ).", "1. Identify coefficients:\n Here, ( a = 1 ), ( b = 1 ), and ( c = -420 ).", "2. Plug into the formula:\n [\n n = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-420)}}{2(1)}\n ]", "3. Simplify inside the square root:\n [\n n = \frac{-1 \pm \sqrt{1 + 1680}}{2}\n ]\n [\n n = \frac{-1 \pm \sqrt{1681}}{2}\n ]", "4. Compute the square root:\n Since ( \sqrt{1681} = 41 ) (because ( 41^2 = 1,681 )),\n [\n n = \frac{-1 \pm 41}{2}\n ]", "5. Solve for both roots:\n - First solution:\n [\n n = \frac{-1 + 41}{2} = \frac{40}{2} = 20\n ]\n - Second solution:\n [\n n = \frac{-1 - 41}{2} = \frac{-42}{2} = -21\n ]", "### Final Solutions", "The two real solutions to the equation ( x^2 + x - 420 = 0 ) are:\n[\nn = 20 \quad \ ext{and} \quad n = -21\n]", "---", "### Why This Method Matters", "The quadratic formula is essential in many disciplines — from physics and engineering to finance and computer science — whenever parabolic relationships place discrete outcomes within reach. Using ( a = 1 ), ( b = 1 ), and ( c = -420 ) simplifies calculations while illustrating how the discriminant (( b^2 - 4ac = 1681 )) drives the nature of the roots.", "In this case, a positive discriminant confirmed two distinct real roots — one positive and one negative — both valuable in modeling real-world phenomena like projectile motion or financial break-even analysis.", "---", "### Quick Recap", "| Step | Result |\n|--------------------------|---------------------------|\n| Quadratic Formula | ( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) |\n| Given values | ( a = 1, b = 1, c = -420 ) |\n| Discriminant | ( b^2 - 4ac = 1 + 1680 = 1681 ) |\n| Roots | ( n = 20 ) and ( n = -21 ) |", "---", "### Additional Tips", "- Always compute the discriminant first to anticipate solution types (real distinct, real repeated, or complex).\n- Simplify intermediate expressions step-by-step to minimize errors.\n- Use exact square roots when possible (as with (\sqrt{1681} = 41)) for precise results.", "Whether you’re solving a textbook problem or applying quadratic modeling in your career, mastering the quadratic formula ensures accuracy and confidence in every calculation."]