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Simplifying 3n2 + 6n = 612 Reorder the terms: 6n + 3n2 = 612 Solving 6n + 3n2 = 612 Solving for variable 'n'. Reorder the terms: -612 + 6n + 3n2 = 612 + -612 Combine like terms: 612 + -612 = 0 -612 + 6n + 3n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-204 + 2n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-204 + 2n + n2)' equal to zero and attempt to solve: Simplifying -204 + 2n + n2 = 0 Solving -204 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '204' to each side of the equation. -204 + 2n + 204 + n2 = 0 + 204 Reorder the terms: -204 + 204 + 2n + n2 = 0 + 204 Combine like terms: -204 + 204 = 0 0 + 2n + n2 = 0 + 204 2n + n2 = 0 + 204 Combine like terms: 0 + 204 = 204 2n + n2 = 204 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 204 + 1 Reorder the terms: 1 + 2n + n2 = 204 + 1 Combine like terms: 204 + 1 = 205 1 + 2n + n2 = 205 Factor a perfect square on the left side: (n + 1)(n + 1) = 205 Calculate the square root of the right side: 14.317821063 Break this problem into two subproblems by setting (n + 1) equal to 14.317821063 and -14.317821063.Subproblem 1
n + 1 = 14.317821063 Simplifying n + 1 = 14.317821063 Reorder the terms: 1 + n = 14.317821063 Solving 1 + n = 14.317821063 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 14.317821063 + -1 Combine like terms: 1 + -1 = 0 0 + n = 14.317821063 + -1 n = 14.317821063 + -1 Combine like terms: 14.317821063 + -1 = 13.317821063 n = 13.317821063 Simplifying n = 13.317821063Subproblem 2
n + 1 = -14.317821063 Simplifying n + 1 = -14.317821063 Reorder the terms: 1 + n = -14.317821063 Solving 1 + n = -14.317821063 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -14.317821063 + -1 Combine like terms: 1 + -1 = 0 0 + n = -14.317821063 + -1 n = -14.317821063 + -1 Combine like terms: -14.317821063 + -1 = -15.317821063 n = -15.317821063 Simplifying n = -15.317821063Solution
The solution to the problem is based on the solutions from the subproblems. n = {13.317821063, -15.317821063}Solution
n = {13.317821063, -15.317821063}
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