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Simplifying 3k2 + -3k + -3 = 0 Reorder the terms: -3 + -3k + 3k2 = 0 Solving -3 + -3k + 3k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + -1k + k2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + -1k + k2)' equal to zero and attempt to solve: Simplifying -1 + -1k + k2 = 0 Solving -1 + -1k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -1k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + -1k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1k + k2 = 0 + 1 -1k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1k + k2 = 1 The k term is -1k. 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. -1k + 0.25 + k2 = 1 + 0.25 Reorder the terms: 0.25 + -1k + k2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1k + k2 = 1.25 Factor a perfect square on the left side: (k + -0.5)(k + -0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (k + -0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
k + -0.5 = 1.118033989 Simplifying k + -0.5 = 1.118033989 Reorder the terms: -0.5 + k = 1.118033989 Solving -0.5 + k = 1.118033989 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + k = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = 1.118033989 + 0.5 k = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 k = 1.618033989 Simplifying k = 1.618033989Subproblem 2
k + -0.5 = -1.118033989 Simplifying k + -0.5 = -1.118033989 Reorder the terms: -0.5 + k = -1.118033989 Solving -0.5 + k = -1.118033989 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + k = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = -1.118033989 + 0.5 k = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 k = -0.618033989 Simplifying k = -0.618033989Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.618033989, -0.618033989}Solution
k = {1.618033989, -0.618033989}
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