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Simplifying 5n(n + -20) + 12 + -9n = 0 Reorder the terms: 5n(-20 + n) + 12 + -9n = 0 (-20 * 5n + n * 5n) + 12 + -9n = 0 (-100n + 5n2) + 12 + -9n = 0 Reorder the terms: 12 + -100n + -9n + 5n2 = 0 Combine like terms: -100n + -9n = -109n 12 + -109n + 5n2 = 0 Solving 12 + -109n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2.4 + -21.8n + n2 = 0 Move the constant term to the right: Add '-2.4' to each side of the equation. 2.4 + -21.8n + -2.4 + n2 = 0 + -2.4 Reorder the terms: 2.4 + -2.4 + -21.8n + n2 = 0 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + -21.8n + n2 = 0 + -2.4 -21.8n + n2 = 0 + -2.4 Combine like terms: 0 + -2.4 = -2.4 -21.8n + n2 = -2.4 The n term is -21.8n. Take half its coefficient (-10.9). Square it (118.81) and add it to both sides. Add '118.81' to each side of the equation. -21.8n + 118.81 + n2 = -2.4 + 118.81 Reorder the terms: 118.81 + -21.8n + n2 = -2.4 + 118.81 Combine like terms: -2.4 + 118.81 = 116.41 118.81 + -21.8n + n2 = 116.41 Factor a perfect square on the left side: (n + -10.9)(n + -10.9) = 116.41 Calculate the square root of the right side: 10.789346597 Break this problem into two subproblems by setting (n + -10.9) equal to 10.789346597 and -10.789346597.Subproblem 1
n + -10.9 = 10.789346597 Simplifying n + -10.9 = 10.789346597 Reorder the terms: -10.9 + n = 10.789346597 Solving -10.9 + n = 10.789346597 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '10.9' to each side of the equation. -10.9 + 10.9 + n = 10.789346597 + 10.9 Combine like terms: -10.9 + 10.9 = 0.0 0.0 + n = 10.789346597 + 10.9 n = 10.789346597 + 10.9 Combine like terms: 10.789346597 + 10.9 = 21.689346597 n = 21.689346597 Simplifying n = 21.689346597Subproblem 2
n + -10.9 = -10.789346597 Simplifying n + -10.9 = -10.789346597 Reorder the terms: -10.9 + n = -10.789346597 Solving -10.9 + n = -10.789346597 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '10.9' to each side of the equation. -10.9 + 10.9 + n = -10.789346597 + 10.9 Combine like terms: -10.9 + 10.9 = 0.0 0.0 + n = -10.789346597 + 10.9 n = -10.789346597 + 10.9 Combine like terms: -10.789346597 + 10.9 = 0.110653403 n = 0.110653403 Simplifying n = 0.110653403Solution
The solution to the problem is based on the solutions from the subproblems. n = {21.689346597, 0.110653403}
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