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Simplifying 2n2 + 18n = -6 Reorder the terms: 18n + 2n2 = -6 Solving 18n + 2n2 = -6 Solving for variable 'n'. Reorder the terms: 6 + 18n + 2n2 = -6 + 6 Combine like terms: -6 + 6 = 0 6 + 18n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(3 + 9n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(3 + 9n + n2)' equal to zero and attempt to solve: Simplifying 3 + 9n + n2 = 0 Solving 3 + 9n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + 9n + -3 + n2 = 0 + -3 Reorder the terms: 3 + -3 + 9n + n2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 9n + n2 = 0 + -3 9n + n2 = 0 + -3 Combine like terms: 0 + -3 = -3 9n + n2 = -3 The n term is 9n. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9n + 20.25 + n2 = -3 + 20.25 Reorder the terms: 20.25 + 9n + n2 = -3 + 20.25 Combine like terms: -3 + 20.25 = 17.25 20.25 + 9n + n2 = 17.25 Factor a perfect square on the left side: (n + 4.5)(n + 4.5) = 17.25 Calculate the square root of the right side: 4.153311931 Break this problem into two subproblems by setting (n + 4.5) equal to 4.153311931 and -4.153311931.Subproblem 1
n + 4.5 = 4.153311931 Simplifying n + 4.5 = 4.153311931 Reorder the terms: 4.5 + n = 4.153311931 Solving 4.5 + n = 4.153311931 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 4.153311931 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 4.153311931 + -4.5 n = 4.153311931 + -4.5 Combine like terms: 4.153311931 + -4.5 = -0.346688069 n = -0.346688069 Simplifying n = -0.346688069Subproblem 2
n + 4.5 = -4.153311931 Simplifying n + 4.5 = -4.153311931 Reorder the terms: 4.5 + n = -4.153311931 Solving 4.5 + n = -4.153311931 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -4.153311931 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -4.153311931 + -4.5 n = -4.153311931 + -4.5 Combine like terms: -4.153311931 + -4.5 = -8.653311931 n = -8.653311931 Simplifying n = -8.653311931Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.346688069, -8.653311931}Solution
n = {-0.346688069, -8.653311931}
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