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Simplifying c3 + c2 + -1c = 0 Reorder the terms: -1c + c2 + c3 = 0 Solving -1c + c2 + c3 = 0 Solving for variable 'c'. Factor out the Greatest Common Factor (GCF), 'c'. c(-1 + c + c2) = 0Subproblem 1
Set the factor 'c' equal to zero and attempt to solve: Simplifying c = 0 Solving c = 0 Move all terms containing c to the left, all other terms to the right. Simplifying c = 0Subproblem 2
Set the factor '(-1 + c + c2)' equal to zero and attempt to solve: Simplifying -1 + c + c2 = 0 Solving -1 + c + c2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + c + 1 + c2 = 0 + 1 Reorder the terms: -1 + 1 + c + c2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + c + c2 = 0 + 1 c + c2 = 0 + 1 Combine like terms: 0 + 1 = 1 c + c2 = 1 The c term is c. 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. c + 0.25 + c2 = 1 + 0.25 Reorder the terms: 0.25 + c + c2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + c + c2 = 1.25 Factor a perfect square on the left side: (c + 0.5)(c + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (c + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
c + 0.5 = 1.118033989 Simplifying c + 0.5 = 1.118033989 Reorder the terms: 0.5 + c = 1.118033989 Solving 0.5 + c = 1.118033989 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + c = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + c = 1.118033989 + -0.5 c = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 c = 0.618033989 Simplifying c = 0.618033989Subproblem 2
c + 0.5 = -1.118033989 Simplifying c + 0.5 = -1.118033989 Reorder the terms: 0.5 + c = -1.118033989 Solving 0.5 + c = -1.118033989 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + c = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + c = -1.118033989 + -0.5 c = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 c = -1.618033989 Simplifying c = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.618033989, -1.618033989}Solution
c = {0, 0.618033989, -1.618033989}
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