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Simplifying 0 = 12n2 + 74n + -910 Reorder the terms: 0 = -910 + 74n + 12n2 Solving 0 = -910 + 74n + 12n2 Solving for variable 'n'. Combine like terms: 0 + 910 = 910 910 + -74n + -12n2 = -910 + 74n + 12n2 + 910 + -74n + -12n2 Reorder the terms: 910 + -74n + -12n2 = -910 + 910 + 74n + -74n + 12n2 + -12n2 Combine like terms: -910 + 910 = 0 910 + -74n + -12n2 = 0 + 74n + -74n + 12n2 + -12n2 910 + -74n + -12n2 = 74n + -74n + 12n2 + -12n2 Combine like terms: 74n + -74n = 0 910 + -74n + -12n2 = 0 + 12n2 + -12n2 910 + -74n + -12n2 = 12n2 + -12n2 Combine like terms: 12n2 + -12n2 = 0 910 + -74n + -12n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(455 + -37n + -6n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(455 + -37n + -6n2)' equal to zero and attempt to solve: Simplifying 455 + -37n + -6n2 = 0 Solving 455 + -37n + -6n2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -75.83333333 + 6.166666667n + n2 = 0 Move the constant term to the right: Add '75.83333333' to each side of the equation. -75.83333333 + 6.166666667n + 75.83333333 + n2 = 0 + 75.83333333 Reorder the terms: -75.83333333 + 75.83333333 + 6.166666667n + n2 = 0 + 75.83333333 Combine like terms: -75.83333333 + 75.83333333 = 0.00000000 0.00000000 + 6.166666667n + n2 = 0 + 75.83333333 6.166666667n + n2 = 0 + 75.83333333 Combine like terms: 0 + 75.83333333 = 75.83333333 6.166666667n + n2 = 75.83333333 The n term is 6.166666667n. Take half its coefficient (3.083333334). Square it (9.506944449) and add it to both sides. Add '9.506944449' to each side of the equation. 6.166666667n + 9.506944449 + n2 = 75.83333333 + 9.506944449 Reorder the terms: 9.506944449 + 6.166666667n + n2 = 75.83333333 + 9.506944449 Combine like terms: 75.83333333 + 9.506944449 = 85.340277779 9.506944449 + 6.166666667n + n2 = 85.340277779 Factor a perfect square on the left side: (n + 3.083333334)(n + 3.083333334) = 85.340277779 Calculate the square root of the right side: 9.237980179 Break this problem into two subproblems by setting (n + 3.083333334) equal to 9.237980179 and -9.237980179.Subproblem 1
n + 3.083333334 = 9.237980179 Simplifying n + 3.083333334 = 9.237980179 Reorder the terms: 3.083333334 + n = 9.237980179 Solving 3.083333334 + n = 9.237980179 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.083333334' to each side of the equation. 3.083333334 + -3.083333334 + n = 9.237980179 + -3.083333334 Combine like terms: 3.083333334 + -3.083333334 = 0.000000000 0.000000000 + n = 9.237980179 + -3.083333334 n = 9.237980179 + -3.083333334 Combine like terms: 9.237980179 + -3.083333334 = 6.154646845 n = 6.154646845 Simplifying n = 6.154646845Subproblem 2
n + 3.083333334 = -9.237980179 Simplifying n + 3.083333334 = -9.237980179 Reorder the terms: 3.083333334 + n = -9.237980179 Solving 3.083333334 + n = -9.237980179 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.083333334' to each side of the equation. 3.083333334 + -3.083333334 + n = -9.237980179 + -3.083333334 Combine like terms: 3.083333334 + -3.083333334 = 0.000000000 0.000000000 + n = -9.237980179 + -3.083333334 n = -9.237980179 + -3.083333334 Combine like terms: -9.237980179 + -3.083333334 = -12.321313513 n = -12.321313513 Simplifying n = -12.321313513Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.154646845, -12.321313513}Solution
n = {6.154646845, -12.321313513}
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