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Simplifying 31000 = 2.6n2 + 7n + -2 Reorder the terms: 31000 = -2 + 7n + 2.6n2 Solving 31000 = -2 + 7n + 2.6n2 Solving for variable 'n'. Combine like terms: 31000 + 2 = 31002 31002 + -7n + -2.6n2 = -2 + 7n + 2.6n2 + 2 + -7n + -2.6n2 Reorder the terms: 31002 + -7n + -2.6n2 = -2 + 2 + 7n + -7n + 2.6n2 + -2.6n2 Combine like terms: -2 + 2 = 0 31002 + -7n + -2.6n2 = 0 + 7n + -7n + 2.6n2 + -2.6n2 31002 + -7n + -2.6n2 = 7n + -7n + 2.6n2 + -2.6n2 Combine like terms: 7n + -7n = 0 31002 + -7n + -2.6n2 = 0 + 2.6n2 + -2.6n2 31002 + -7n + -2.6n2 = 2.6n2 + -2.6n2 Combine like terms: 2.6n2 + -2.6n2 = 0.0 31002 + -7n + -2.6n2 = 0.0 Begin completing the square. Divide all terms by -2.6 the coefficient of the squared term: Divide each side by '-2.6'. -11923.84615 + 2.692307692n + n2 = 0 Move the constant term to the right: Add '11923.84615' to each side of the equation. -11923.84615 + 2.692307692n + 11923.84615 + n2 = 0 + 11923.84615 Reorder the terms: -11923.84615 + 11923.84615 + 2.692307692n + n2 = 0 + 11923.84615 Combine like terms: -11923.84615 + 11923.84615 = 0.00000 0.00000 + 2.692307692n + n2 = 0 + 11923.84615 2.692307692n + n2 = 0 + 11923.84615 Combine like terms: 0 + 11923.84615 = 11923.84615 2.692307692n + n2 = 11923.84615 The n term is 2.692307692n. Take half its coefficient (1.346153846). Square it (1.812130177) and add it to both sides. Add '1.812130177' to each side of the equation. 2.692307692n + 1.812130177 + n2 = 11923.84615 + 1.812130177 Reorder the terms: 1.812130177 + 2.692307692n + n2 = 11923.84615 + 1.812130177 Combine like terms: 11923.84615 + 1.812130177 = 11925.658280177 1.812130177 + 2.692307692n + n2 = 11925.658280177 Factor a perfect square on the left side: (n + 1.346153846)(n + 1.346153846) = 11925.658280177 Calculate the square root of the right side: 109.204662355 Break this problem into two subproblems by setting (n + 1.346153846) equal to 109.204662355 and -109.204662355.Subproblem 1
n + 1.346153846 = 109.204662355 Simplifying n + 1.346153846 = 109.204662355 Reorder the terms: 1.346153846 + n = 109.204662355 Solving 1.346153846 + n = 109.204662355 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.346153846' to each side of the equation. 1.346153846 + -1.346153846 + n = 109.204662355 + -1.346153846 Combine like terms: 1.346153846 + -1.346153846 = 0.000000000 0.000000000 + n = 109.204662355 + -1.346153846 n = 109.204662355 + -1.346153846 Combine like terms: 109.204662355 + -1.346153846 = 107.858508509 n = 107.858508509 Simplifying n = 107.858508509Subproblem 2
n + 1.346153846 = -109.204662355 Simplifying n + 1.346153846 = -109.204662355 Reorder the terms: 1.346153846 + n = -109.204662355 Solving 1.346153846 + n = -109.204662355 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.346153846' to each side of the equation. 1.346153846 + -1.346153846 + n = -109.204662355 + -1.346153846 Combine like terms: 1.346153846 + -1.346153846 = 0.000000000 0.000000000 + n = -109.204662355 + -1.346153846 n = -109.204662355 + -1.346153846 Combine like terms: -109.204662355 + -1.346153846 = -110.550816201 n = -110.550816201 Simplifying n = -110.550816201Solution
The solution to the problem is based on the solutions from the subproblems. n = {107.858508509, -110.550816201}
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