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Simplifying -2 + -13.8n = 8n(6n + 1) Reorder the terms: -2 + -13.8n = 8n(1 + 6n) -2 + -13.8n = (1 * 8n + 6n * 8n) -2 + -13.8n = (8n + 48n2) Solving -2 + -13.8n = 8n + 48n2 Solving for variable 'n'. Combine like terms: -13.8n + -8n = -21.8n -2 + -21.8n + -48n2 = 8n + 48n2 + -8n + -48n2 Reorder the terms: -2 + -21.8n + -48n2 = 8n + -8n + 48n2 + -48n2 Combine like terms: 8n + -8n = 0 -2 + -21.8n + -48n2 = 0 + 48n2 + -48n2 -2 + -21.8n + -48n2 = 48n2 + -48n2 Combine like terms: 48n2 + -48n2 = 0 -2 + -21.8n + -48n2 = 0 Begin completing the square. Divide all terms by -48 the coefficient of the squared term: Divide each side by '-48'. 0.04166666667 + 0.4541666667n + n2 = 0 Move the constant term to the right: Add '-0.04166666667' to each side of the equation. 0.04166666667 + 0.4541666667n + -0.04166666667 + n2 = 0 + -0.04166666667 Reorder the terms: 0.04166666667 + -0.04166666667 + 0.4541666667n + n2 = 0 + -0.04166666667 Combine like terms: 0.04166666667 + -0.04166666667 = 0.00000000000 0.00000000000 + 0.4541666667n + n2 = 0 + -0.04166666667 0.4541666667n + n2 = 0 + -0.04166666667 Combine like terms: 0 + -0.04166666667 = -0.04166666667 0.4541666667n + n2 = -0.04166666667 The n term is 0.4541666667n. Take half its coefficient (0.2270833334). Square it (0.05156684031) and add it to both sides. Add '0.05156684031' to each side of the equation. 0.4541666667n + 0.05156684031 + n2 = -0.04166666667 + 0.05156684031 Reorder the terms: 0.05156684031 + 0.4541666667n + n2 = -0.04166666667 + 0.05156684031 Combine like terms: -0.04166666667 + 0.05156684031 = 0.00990017364 0.05156684031 + 0.4541666667n + n2 = 0.00990017364 Factor a perfect square on the left side: (n + 0.2270833334)(n + 0.2270833334) = 0.00990017364 Calculate the square root of the right side: 0.099499616 Break this problem into two subproblems by setting (n + 0.2270833334) equal to 0.099499616 and -0.099499616.Subproblem 1
n + 0.2270833334 = 0.099499616 Simplifying n + 0.2270833334 = 0.099499616 Reorder the terms: 0.2270833334 + n = 0.099499616 Solving 0.2270833334 + n = 0.099499616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2270833334' to each side of the equation. 0.2270833334 + -0.2270833334 + n = 0.099499616 + -0.2270833334 Combine like terms: 0.2270833334 + -0.2270833334 = 0.0000000000 0.0000000000 + n = 0.099499616 + -0.2270833334 n = 0.099499616 + -0.2270833334 Combine like terms: 0.099499616 + -0.2270833334 = -0.1275837174 n = -0.1275837174 Simplifying n = -0.1275837174Subproblem 2
n + 0.2270833334 = -0.099499616 Simplifying n + 0.2270833334 = -0.099499616 Reorder the terms: 0.2270833334 + n = -0.099499616 Solving 0.2270833334 + n = -0.099499616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2270833334' to each side of the equation. 0.2270833334 + -0.2270833334 + n = -0.099499616 + -0.2270833334 Combine like terms: 0.2270833334 + -0.2270833334 = 0.0000000000 0.0000000000 + n = -0.099499616 + -0.2270833334 n = -0.099499616 + -0.2270833334 Combine like terms: -0.099499616 + -0.2270833334 = -0.3265829494 n = -0.3265829494 Simplifying n = -0.3265829494Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.1275837174, -0.3265829494}
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