So \( 2n^2 + 2n = 120 → n^2 + n − 60 = 0 \)

["Solving the Quadratic Equation: How to Solve ( 2n^2 + 2n = 120 ) Step-by-Step", "Understanding how to solve quadratic equations is essential for students, engineers, and anyone working with mathematical modeling. One common problem is simplifying and solving equations of the form ( 2n^2 + 2n = 120 ), which leads to the quadratic equation ( n^2 + n − 60 = 0 ). In this SEO-optimized guide, we’ll break down the solution process clearly, explain each step, and show how to efficiently solve ( n^2 + n − 60 = 0 ) — making it easier for learners and professionals alike.", "---", "### From Word Problem to Quadratic Equation\nLet’s start with the real-world equation:\n[ 2n^2 + 2n = 120 ]", "To solve it, first move all terms to one side:\n[ 2n^2 + 2n − 120 = 0 ]", "Divide every term by 2 to simplify:\n[ n^2 + n − 60 = 0 ]", "Now we have a standard quadratic equation:\n[ n^2 + n − 60 = 0 ]", "---", "### Solving ( n^2 + n − 60 = 0 ) Efficiently\nQuadratic equations in the form ( ax^2 + bx + c = 0 ) can be solved using:", "- Factoring (if factors exist),\n- Quadratic formula,\n- Completing the square.", "Let’s explore factoring first since the numbers here are manageable.", "#### Step 1: Factoring the Quadratic\nWe look for two numbers that:\n- Multiply to ( c = -60 )\n- Add to ( b = 1 )", "After testing factor pairs of −60, we find:\n6 and −10 satisfy:\n( 6 \ imes (−10) = −60 )\n( 6 + (−10) = −4 ) → Not correct", "Try −5 and 12:\n( -5 \ imes 12 = −60 )\n( -5 + 12 = 7 ) → Not correct", "Try −4 and 15:\n( -4 \ imes 15 = −60 )\n( -4 + 15 = 11 ) → Not correct", "Try −6 and 10:\n( -6 \ imes 10 = −60 )\n( -6 + 10 = 4 ) → Closer, but not −1", "Try −8 and 7.5 → Not integer factors", "Try 5 and −12:\n( 5 \ imes (−12) = −60 ), sum = −7 → nope", "Wait — try −5 and 12 again? Sum is 7. Try little different:", "Try −5 × 12 = −60 → sum = 7\nBut we need sum = +1", "Try −5 + 12 = +7. What if we reverse signs?", "Try −8 and 7.5? Not clean.", "Try −4 and 15? Sum = 11\nTry −3 and 20? Sum = 17 → too big", "Wait — let’s check: is there a pair such that\n( p \ imes q = -60 ) and ( p + q = 1 )", "Try −8 and 7.5? No", "Wait — try −5 and 12 sum 7\nTry −4 and −15 → sum −19\nTry −5 and 12 → 7\nTry −6 and 10 → 4\nTry −10 and 6 → −4", "We are stuck? Wait — try −5 and 12 no\nWait — try −8 and 7.5? Not integers", "Wait — perhaps −4 and −15? Sum −19, product +60", "Wait — let’s use the factoring method systematically.", "We need two integers ( m ) and ( n ) such that:\n[\n(n + m)(n - k) \quad \ ext{such that } m \cdot (-k) = -60, ; m - k = 1\n]\nActually, better: factor as ( (n + a)(n + b) = 0 ), where ( a \cdot b = -60 ), ( a + b = 1 )", "Try all factor pairs of −60:", "| ( a ) | ( b ) | ( a + b ) |\n|--------|---------|-------------|\n| −60 | 1 | −59 |\n| −30 | 2 | −28 |\n| −20 | 3 | −17 |\n| −15 | 4 | −11 |\n| −12 | 5 | −7 |\n| −10 | 6 | −4 |\n| −8 | 7.5 → no | — |\n| 6 | −10 | −4 |\n| 10 | −6 | 4 |\n| 12 | −5 | 7 |\n| 15 | −4 | 11 |\n| 20 | −3 | 17 |\n| 30 | −2 | 28 |\n| 60 | −1 | 59 |", "No pair adds to 1. So factoring is not straightforward with integers.", "---", "### Step 2: Use the Quadratic Formula\nSince factoring is difficult, apply the quadratic formula:\n[\nn = \frac{−b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For ( n^2 + n − 60 = 0 ),\n( a = 1 ), ( b = 1 ), ( c = −60 )", "Compute discriminant:\n[\n\Delta = b^2 - 4ac = 1^2 - 4(1)(−60) = 1 + 240 = 241\n]", "Since ( \sqrt{241} ) is irrational, the solutions are:\n[\nn = \frac{−1 \pm \sqrt{241}}{2}\n]", "Thus, the two real solutions are:\n[\nn = \frac{−1 + \sqrt{241}}{2} \quad \ ext{and} \quad n = \frac{−1 - \sqrt{241}}{2}\n]", "---", "### Step 3: Estimating Solutions for Practical Use\nSince ( \sqrt{241} \approx 15.52 ),\n[\nn \approx \frac{−1 + 15.52}{2} = \frac{14.52}{2} = 7.26\n]\n[\nn \approx \frac{−1 - 15.52}{2} = \frac{−16.52}{2} = -8.26\n]", "Since ( n ) often represents a count or measurable quantity, we typically take the positive root:\n[\nn \approx 7.26\n]", "For exact purposes, we keep the precise form.", "---", "### Applications and Why This Equation Matters\nEquations like ( 2n^2 + 2n = 120 ) appear in:\n- Physics (motion under constant acceleration),\n- Economics (revenue and cost modeling),\n- Computer science (time complexity of nested loops).", "Simplifying to ( n^2 + n − 60 = 0 ) enables efficient analysis and solution.", "---", "### Final Thoughts\nSolving ( 2n^2 + 2n = 120 ) leads neatly to the quadratic equation ( n^2 + n − 60 = 0 ). While factoring is tricky here, the quadratic formula provides a direct path to accurate solutions. Whether you're a student tutoring others or a professional applying math, mastering these techniques ensures fluency in algebra.", "---", "Key takeaways:\n- Simplify quadratic equations before solving.\n- When factoring is difficult, use the quadratic formula.\n- Real-world problems often hide elegant math — always write full steps.", "For more algebra help, explore related topics like the discriminant, rational root theorem, or completing the square.", "---", "Keywords: solve (2n^2 + 2n = 120), quadratic equation (n^2 + n − 60 = 0), quadratic formula, factoring quadratics, algebra tips, step-by-step solution, solving quadratic equations, simplified quadratic methods.", "---", "Meta Description: Step-by-step guide to solving (2n^2 + 2n = 120) by reducing to (n^2 + n − 60 = 0), using the quadratic formula with exact values and real-world applications. Perfect for students and educators."]