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Simplifying 3n2 + -3 = 3n Reorder the terms: -3 + 3n2 = 3n Solving -3 + 3n2 = 3n Solving for variable 'n'. Reorder the terms: -3 + -3n + 3n2 = 3n + -3n Combine like terms: 3n + -3n = 0 -3 + -3n + 3n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + -1n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + -1n + n2)' equal to zero and attempt to solve: Simplifying -1 + -1n + n2 = 0 Solving -1 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -1n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -1n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1n + n2 = 0 + 1 -1n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1n + n2 = 1 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 1 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1n + n2 = 1.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (n + -0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
n + -0.5 = 1.118033989 Simplifying n + -0.5 = 1.118033989 Reorder the terms: -0.5 + n = 1.118033989 Solving -0.5 + n = 1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.118033989 + 0.5 n = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 n = 1.618033989 Simplifying n = 1.618033989Subproblem 2
n + -0.5 = -1.118033989 Simplifying n + -0.5 = -1.118033989 Reorder the terms: -0.5 + n = -1.118033989 Solving -0.5 + n = -1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.118033989 + 0.5 n = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 n = -0.618033989 Simplifying n = -0.618033989Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.618033989, -0.618033989}Solution
n = {1.618033989, -0.618033989}
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