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Simplifying 6n2 + 2n + -1000 = 0 Reorder the terms: -1000 + 2n + 6n2 = 0 Solving -1000 + 2n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-500 + n + 3n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-500 + n + 3n2)' equal to zero and attempt to solve: Simplifying -500 + n + 3n2 = 0 Solving -500 + n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -166.6666667 + 0.3333333333n + n2 = 0 Move the constant term to the right: Add '166.6666667' to each side of the equation. -166.6666667 + 0.3333333333n + 166.6666667 + n2 = 0 + 166.6666667 Reorder the terms: -166.6666667 + 166.6666667 + 0.3333333333n + n2 = 0 + 166.6666667 Combine like terms: -166.6666667 + 166.6666667 = 0.0000000 0.0000000 + 0.3333333333n + n2 = 0 + 166.6666667 0.3333333333n + n2 = 0 + 166.6666667 Combine like terms: 0 + 166.6666667 = 166.6666667 0.3333333333n + n2 = 166.6666667 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 = 166.6666667 + 0.25 Reorder the terms: 0.25 + 0.3333333333n + n2 = 166.6666667 + 0.25 Combine like terms: 166.6666667 + 0.25 = 166.9166667 0.25 + 0.3333333333n + n2 = 166.9166667 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 166.9166667 Calculate the square root of the right side: 12.919623319 Break this problem into two subproblems by setting (n + 0.5) equal to 12.919623319 and -12.919623319.Subproblem 1
n + 0.5 = 12.919623319 Simplifying n + 0.5 = 12.919623319 Reorder the terms: 0.5 + n = 12.919623319 Solving 0.5 + n = 12.919623319 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 = 12.919623319 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 12.919623319 + -0.5 n = 12.919623319 + -0.5 Combine like terms: 12.919623319 + -0.5 = 12.419623319 n = 12.419623319 Simplifying n = 12.419623319Subproblem 2
n + 0.5 = -12.919623319 Simplifying n + 0.5 = -12.919623319 Reorder the terms: 0.5 + n = -12.919623319 Solving 0.5 + n = -12.919623319 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 = -12.919623319 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -12.919623319 + -0.5 n = -12.919623319 + -0.5 Combine like terms: -12.919623319 + -0.5 = -13.419623319 n = -13.419623319 Simplifying n = -13.419623319Solution
The solution to the problem is based on the solutions from the subproblems. n = {12.419623319, -13.419623319}Solution
n = {12.419623319, -13.419623319}
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