n^2 + 2n - 105 = 0

["# Solving the Quadratic Equation n² + 2n – 105 = 0: A Step-by-Step Guide", "When encountering a quadratic equation like n² + 2n – 105 = 0, solving it efficiently is key—especially for students, educators, and self-learners looking to master algebra. This article walks you through solving the equation step-by-step and explains its real-world applications and importance in mathematics.", "---", "## What is n² + 2n – 105 = 0?", "The equation n² + 2n – 105 = 0 is a standard quadratic equation in two variables:\nax² + bx + c = 0, where:\n- a = 1\n- b = 2\n- c = –105", "Quadratic equations are important in algebra because they model various phenomena, from projectile motion in physics to economic growth patterns in business. Understanding how to solve them unlocks deeper insights into mathematical reasoning and problem-solving.", "---", "## Why Solve n² + 2n – 105 = 0?", "This equation serves as a classic example for:\n- Learning how to factor quadratic expressions\n- Applying the quadratic formula\n- Verifying solutions via substitution\n- Applying algebra to real-life scenarios like area problems, optimization, and engineering calculations", "---", "## Step 1: Choose a Solution Method", "For n² + 2n – 105 = 0, two proven approaches exist:", "1. Factoring – Solve by expressing the equation as a product of two binomials.\n2. Quadratic Formula – Use the reliable general solution for any quadratic equation.", "---", "## Step 2: Solving by Factoring (Efficient if possible)", "We look for two numbers that:\n- Multiply to c = –105\n- Add to b = 2", "After checking factor pairs of -105, we find:\n10 and –15\nbecause:\n10 × (–15) = –105\n10 + (–15) = –5 → Not correct!", "Wait — correction:\nWe need two numbers that sum to +2 and multiply to –105.", "Try: 15 and –7\n15 × (–7) = –105\n15 + (–7) = 8 → No", "Try: 15 and –7 failed — how about 15 and –7 is too high.", "Try: –7 and 15? No.", "Correct pair: –10 and 15? No.", "Wait — let’s list factor pairs of 105:\n1 × 105\n3 × 35\n5 × 21\n7 × 15", "Now try combinations with opposite signs:", "Try: 15 and –7? Sum = 8\nTry: –15 and 7? Sum = –8\nTry: –10 and 10.5? Not integer", "Wait — half of 105 is 21, so try:\n–15 and 7: sum = –8\n–7 and 15: sum = +8\n–5 and 21: sum = +16\n–3 and 35: sum = +32\n–1 and 105: sum = +104", "None sum to +2?", "Wait — perhaps factoring isn’t straightforward. Let’s verify the discriminant first.", "---", "## Step 3: Use the Quadratic Formula (Always Reliable)", "The quadratic formula is:\nn = [–b ± √(b² – 4ac)] / (2a)", "For n² + 2n – 105 = 0:\na = 1, b = 2, c = –105", "### Step 3.1: Compute the discriminant D = b² – 4ac\nD = (2)² – 4(1)(–105)\n= 4 + 420\n= 424", "√424 simplifies:\n424 = 4 × 106 = 4 × 2 × 53 → √424 = 2√106", "So,\nn = [–2 ± √424] / 2\n= [–2 ± 2√106] / 2\n= –1 ± √106", "Final solutions:\nn = –1 + √106\nn = –1 – √106", "---", "## Step 4: Approximate Numerical Values (Optional but Helpful)", "Since √106 ≈ 10.30,\nn ≈ –1 + 10.30 = 9.30\nand\nn ≈ –1 – 10.30 = –11.30", "---", "## Step 5: Verify the Solutions", "Plug n ≈ 9.30 into original equation:\n(9.30)² + 2(9.30) – 105 ≈ 86.49 + 18.6 – 105 = 105.09 – 105 ≈ 0.09 → Close (error due to rounding)", "Exact solution confirms validity.", "---", "## Real-World Applications", "Equations like n² + 2n – 105 = 0 can model:", "- Projectile motion: Finding time when height equals a target value\n- Area problems: Finding dimensions of a rectangle with given area and perimeter\n- Business models: Revenue or cost optimization\n- Engineering design: Calculating dimensions for stability or material strength", "---", "## Frequently Asked Questions", "### Q: Can this equation be factored easily?\nSometimes yes, but not always. In this case, with discriminant not a perfect square, factoring is messy. The quadratic formula is preferred.", "### Q: What if c = +105?\nThen the equation would be n² + 2n + 105 = 0 → discriminant = 4 – 420 = –416 → no real solutions", "### Q: How do I improve my factoring skills?\nPractice identifying factor pairs, use trial and error, and expand binomials regularly.", "---", "## Summary", "The quadratic equation n² + 2n – 105 = 0 demonstrates how algebra combines factoring and formulas to solve real-world problems. Beating factoring limits, the discriminant method ensures accurate solutions — even without perfect integer roots.", "Key Takeaways:\n✅ Use the quadratic formula when factoring is complex\n✅ Verify solutions by substitution\n✅ Understand applications in science, engineering, business\n✅ Practice factoring, completing the square, and numerical approximation", "---", "## Want to Solve More Quadratics?", "Visit our Comprehensive Quadratic Equation Solver or try partial factoring guides, discriminant analysis tips, and step-by-step video tutorials.", "---", "Keywords for SEO: quadratic equation solution, n² + 2n – 105 = 0, solve quadratic equation, quadratic formula, factoring quadratics, discriminant calculation, algebra practice, real-world quadratic applications, step-by-step quadratic solving.", "---", "Effortlessly conquer quadratics—understand, solve, and apply!"]