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Simplifying 3n2 + n + -318 = 0 Reorder the terms: -318 + n + 3n2 = 0 Solving -318 + n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -106 + 0.3333333333n + n2 = 0 Move the constant term to the right: Add '106' to each side of the equation. -106 + 0.3333333333n + 106 + n2 = 0 + 106 Reorder the terms: -106 + 106 + 0.3333333333n + n2 = 0 + 106 Combine like terms: -106 + 106 = 0 0 + 0.3333333333n + n2 = 0 + 106 0.3333333333n + n2 = 0 + 106 Combine like terms: 0 + 106 = 106 0.3333333333n + n2 = 106 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333n + 0.25 + n2 = 106 + 0.25 Reorder the terms: 0.25 + 0.3333333333n + n2 = 106 + 0.25 Combine like terms: 106 + 0.25 = 106.25 0.25 + 0.3333333333n + n2 = 106.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 106.25 Calculate the square root of the right side: 10.307764064 Break this problem into two subproblems by setting (n + 0.5) equal to 10.307764064 and -10.307764064.Subproblem 1
n + 0.5 = 10.307764064 Simplifying n + 0.5 = 10.307764064 Reorder the terms: 0.5 + n = 10.307764064 Solving 0.5 + n = 10.307764064 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 10.307764064 + -0.5 n = 10.307764064 + -0.5 Combine like terms: 10.307764064 + -0.5 = 9.807764064 n = 9.807764064 Simplifying n = 9.807764064Subproblem 2
n + 0.5 = -10.307764064 Simplifying n + 0.5 = -10.307764064 Reorder the terms: 0.5 + n = -10.307764064 Solving 0.5 + n = -10.307764064 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -10.307764064 + -0.5 n = -10.307764064 + -0.5 Combine like terms: -10.307764064 + -0.5 = -10.807764064 n = -10.807764064 Simplifying n = -10.807764064Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.807764064, -10.807764064}
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