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Simplifying (-12 + -7n2 + -8n) + (-6n + 12n2 + -13) = 0 Reorder the terms: (-12 + -8n + -7n2) + (-6n + 12n2 + -13) = 0 Remove parenthesis around (-12 + -8n + -7n2) -12 + -8n + -7n2 + (-6n + 12n2 + -13) = 0 Reorder the terms: -12 + -8n + -7n2 + (-13 + -6n + 12n2) = 0 Remove parenthesis around (-13 + -6n + 12n2) -12 + -8n + -7n2 + -13 + -6n + 12n2 = 0 Reorder the terms: -12 + -13 + -8n + -6n + -7n2 + 12n2 = 0 Combine like terms: -12 + -13 = -25 -25 + -8n + -6n + -7n2 + 12n2 = 0 Combine like terms: -8n + -6n = -14n -25 + -14n + -7n2 + 12n2 = 0 Combine like terms: -7n2 + 12n2 = 5n2 -25 + -14n + 5n2 = 0 Solving -25 + -14n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -5 + -2.8n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.8n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -2.8n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.8n + n2 = 0 + 5 -2.8n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.8n + n2 = 5 The n term is -2.8n. Take half its coefficient (-1.4). Square it (1.96) and add it to both sides. Add '1.96' to each side of the equation. -2.8n + 1.96 + n2 = 5 + 1.96 Reorder the terms: 1.96 + -2.8n + n2 = 5 + 1.96 Combine like terms: 5 + 1.96 = 6.96 1.96 + -2.8n + n2 = 6.96 Factor a perfect square on the left side: (n + -1.4)(n + -1.4) = 6.96 Calculate the square root of the right side: 2.638181192 Break this problem into two subproblems by setting (n + -1.4) equal to 2.638181192 and -2.638181192.Subproblem 1
n + -1.4 = 2.638181192 Simplifying n + -1.4 = 2.638181192 Reorder the terms: -1.4 + n = 2.638181192 Solving -1.4 + n = 2.638181192 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.4' to each side of the equation. -1.4 + 1.4 + n = 2.638181192 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + n = 2.638181192 + 1.4 n = 2.638181192 + 1.4 Combine like terms: 2.638181192 + 1.4 = 4.038181192 n = 4.038181192 Simplifying n = 4.038181192Subproblem 2
n + -1.4 = -2.638181192 Simplifying n + -1.4 = -2.638181192 Reorder the terms: -1.4 + n = -2.638181192 Solving -1.4 + n = -2.638181192 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.4' to each side of the equation. -1.4 + 1.4 + n = -2.638181192 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + n = -2.638181192 + 1.4 n = -2.638181192 + 1.4 Combine like terms: -2.638181192 + 1.4 = -1.238181192 n = -1.238181192 Simplifying n = -1.238181192Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.038181192, -1.238181192}
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