\( 3n^2 + 7n - 220 = 0 \)

["# Solving the Quadratic Equation: ( 3n^2 + 7n - 220 = 0 )", "Quadratic equations are essential in algebra and appear in various fields such as physics, engineering, economics, and computer science. The equation ( 3n^2 + 7n - 220 = 0 ) is a standard quadratic that can be solved using multiple methods, including factoring, completing the square, and the quadratic formula. In this article, we’ll explore how to solve this equation completely, why it matters, and how it fits into broader mathematical applications.", "---", "## Understanding the Equation", "The equation ( 3n^2 + 7n - 220 = 0 ) is a quadratic equation in standard form:", "[\nan^2 + bn + c = 0\n]", "Where:\n- ( a = 3 )\n- ( b = 7 )\n- ( c = -220 )", "---", "## Why Solve This Equation?", "Quadratic equations like this one model real-world phenomena:", "- Projectile motion: Finding the time when a projectile hits the ground.\n- Engineering design: Optimizing structural dimensions.\n- Financial calculations: Considering profit or break-even points.\n- Computer science: Algorithm complexity analysis involving polynomial time.", "---", "## Methods to Solve ( 3n^2 + 7n - 220 = 0 )", "### 1. Using the Quadratic Formula", "The quadratic formula is reliable for solving any equation of the form ( ax^2 + bx + c = 0 ):", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging in ( a = 3 ), ( b = 7 ), ( c = -220 ):", "[\nn = \frac{-7 \pm \sqrt{7^2 - 4(3)(-220)}}{2(3)}\n]", "[\nn = \frac{-7 \pm \sqrt{49 + 2640}}{6}\n]\n[\nn = \frac{-7 \pm \sqrt{2689}}{6}\n]", "Since ( 2689 ) is not a perfect square (( 51^2 = 2601 ), ( 52^2 = 2704 )), we approximate:", "[\n\sqrt{2689} \approx 51.86\n]", "So:", "[\nn = \frac{-7 \pm 51.86}{6}\n]", "Compute the two solutions:", "1. ( n = \frac{-7 + 51.86}{6} = \frac{44.86}{6} \approx 7.477 )\n2. ( n = \frac{-7 - 51.86}{6} = \frac{-58.86}{6} \approx -9.81 )", "---", "### 2. Factoring (if applicable)", "Factoring tries to express ( 3n^2 + 7n - 220 ) as a product of two binomials. We look for two numbers multiplying to ( 3 \ imes (-220) = -660 ) and adding to ( 7 ). Testing combinations:", "- ( 30 \ imes (-22) = -660 ) and ( 30 + (-22) = 8 ) → close\n- ( 33 \ imes (-20) = -660 ), sum ( +13 ) → too high\n- ( 33 ) and ( -20 ) — we’re near, but doesn’t work", "Trying adjustment by splitting the middle term:", "[\n3n^2 + 30n - 23n - 220\n]\n[\n= 3n(n + 10) - 23(n + 10)\n]\n[\n= (3n - 23)(n + 10)\n]", "So the factored form is:", "[\n(3n - 23)(n + 10) = 0\n]", "Set each factor to zero:", "- ( 3n - 23 = 0 ) → ( n = \frac{23}{3} \approx 7.67 )\n- ( n + 10 = 0 ) → ( n = -10 )", "Wait — discrepancy! From the quadratic formula, one root was ~7.477, and exact factoring here gives ( n = \frac{23}{3} \approx 7.666... ). The earlier decimal approximation was close but not exact. This confirms that exact rational solutions may come from factoring, while decimals emerge from irrational results when discriminant isn’t a perfect square.", "Indeed, because ( \sqrt{2689} ) is irrational, the exact solution is:", "[\nn = \frac{23}{3}, \quad n = -10\n]", "But wait — confirmation check:", "Plug ( n = \frac{23}{3} \approx 7.6667 ):", "[\n3\left(\frac{23}{3}\right)^2 + 7\left(\frac{23}{3}\right) = 3 \cdot \frac{529}{9} + \frac{161}{3} = \frac{1587}{9} + \frac{483}{9} = \frac{2070}{9} = 230\n]\nThen ( 230 - 220 = 10 ) → wait, inconsistency!", "Recheck factoring:\nWe had:\n( 3n^2 + 7n - 220 )", "Try factoring again carefully:", "We want two binomials:\n( (3n - a)(n + b) = 3n^2 + (3b - a)n - ab )", "Set:", "- ( 3b - a = 7 )\n- ( ab = 220 )", "Try integer factor pairs of 220:\n( (1,220), (2,110), (4,55), (5,44), (10,22) ), etc.", "Try ( a = 30 ), ( b = 22 ):\n( 3(22) - 30 = 66 - 30 = 36 ) → too high\n( a = 55 ), ( b = 4 ): ( 3(4) - 55 = 12 - 55 = -43 ) → no", "Try ( a = 33 ), ( b = 20 ): ( 3(20) - 33 = 60 - 33 = 27 ) → no", "Try ( a = 44 ), ( b = 5 ): ( 15 - 44 = -29 )", "No integer solution, so factoring isn’t straightforward. However, earlier split:", "[\n3n^2 + 30n - 23n - 220 = 3n(n + 10) -23(n + 10) = (3n - 23)(n + 10)\n]", "Yes, correct. So:", "[\n(3n - 23)(n + 10) = 0\n]\n[\n\Rightarrow n = \frac{23}{3} \quad \ ext{or} \quad n = -10\n]", "Check ( n = \frac{23}{3} ):", "Compute ( 3n^2 + 7n ):", "[\n3 \left(\frac{529}{9}\right) + 7 \cdot \frac{23}{3} = \frac{1587}{9} + \frac{161}{3} = \frac{1587 + 483}{9} = \frac{2070}{9} = 230\n]", "Then ( 230 - 220 = 10 <br/>\neq 0 ) → mistake!", "Wait — this means expansion:", "[\n(3n - 23)(n + 10) = 3n^2 + 30n - 23n - 230 = 3n^2 + 7n - 230\n]", "But original equation: ( 3n^2 + 7n - 220 = 0 ), so constant term is off by 10.", "So error in factoring!", "Correct factoring attempt with correct constant:\nWe seek: ( (3n + a)(n + b) = 3n^2 + (3b + a)n + ab )? No — standard is ( (pn + q)(rn + s) ), but better guide earlier split.", "From earlier:", "[\n3n^2 + 7n - 220 = 3n^2 + 30n - 23n - 220 = \ ext{wrong constant}\n]", "Correct split:\nWe want ( ab = -220 ), ( a + 3b = 7 ) (coefficients)", "Try ( b = -10 ): then ( a = 7 - 3(-10) = 7 + 30 = 37 ), but ( ab = 37 \ imes (-10) = -370 <br/>\ne -220 )", "Try ( b = -5 ): ( a = 7 - 3(-5) = 22 ), ( ab = 22 \ imes (-5) = -110 )", "Try ( b = -11 ): ( a = 7 + 33 = 40 ), ( ab = -440 )", "Try ( b = -20 ): ( a = 7 + 60 = 67 ), ( ab = -1340 ) — too big", "Wait — earlier step had arithmetic error.", "Go back:\nWe had:", "[\n3n^2 + 7n - 220 = 3n^2 + (30n - 23n) - 220\n]", "Only if ( ab = -220 ), ( a + 3b = 7 )", "Let ( a = 7 - 3b ), plug into ( ab = -220 ):", "[\n(7 - 3b)b = -220\n]\n[\n7b - 3b^2 = -220\n]\n[\n-3b^2 + 7b + 220 = 0\n]\n[\n3b^2 - 7b - 220 = 0\n]", "Now solve:", "[\nb = \frac{7 \pm \sqrt{49 + 2640}}{6} = \frac{7 \pm \sqrt{2689}}{6}\n]", "Again irrational. So this quadratic does not factor nicely over integers. Thus, quadratic formula is the most reliable method.", "Use:", "[\nn = \frac{-7 \pm \sqrt{7^2 - 4(3)(-220)}}{2(3)} = \frac{-7 \pm \sqrt{2689}}{6}\n]", "Since ( 51^2 = 2601 ), ( 52^2 = 2704 ), ( \sqrt{2689} \approx 51.86 )", "So approximate solutions:", "- ( n = \frac{-7 + 51.86}{6} \approx \frac{44.86}{6} \approx 7.477 )\n- ( n = \frac{-7 - 51.86}{6} \approx \frac{-58.86}{6} \approx -"]