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Simplifying (4c + 3) * 6c = 92 Reorder the terms: (3 + 4c) * 6c = 92 Reorder the terms for easier multiplication: 6c(3 + 4c) = 92 (3 * 6c + 4c * 6c) = 92 (18c + 24c2) = 92 Solving 18c + 24c2 = 92 Solving for variable 'c'. Reorder the terms: -92 + 18c + 24c2 = 92 + -92 Combine like terms: 92 + -92 = 0 -92 + 18c + 24c2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-46 + 9c + 12c2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-46 + 9c + 12c2)' equal to zero and attempt to solve: Simplifying -46 + 9c + 12c2 = 0 Solving -46 + 9c + 12c2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -3.833333333 + 0.75c + c2 = 0 Move the constant term to the right: Add '3.833333333' to each side of the equation. -3.833333333 + 0.75c + 3.833333333 + c2 = 0 + 3.833333333 Reorder the terms: -3.833333333 + 3.833333333 + 0.75c + c2 = 0 + 3.833333333 Combine like terms: -3.833333333 + 3.833333333 = 0.000000000 0.000000000 + 0.75c + c2 = 0 + 3.833333333 0.75c + c2 = 0 + 3.833333333 Combine like terms: 0 + 3.833333333 = 3.833333333 0.75c + c2 = 3.833333333 The c term is 0.75c. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75c + 0.140625 + c2 = 3.833333333 + 0.140625 Reorder the terms: 0.140625 + 0.75c + c2 = 3.833333333 + 0.140625 Combine like terms: 3.833333333 + 0.140625 = 3.973958333 0.140625 + 0.75c + c2 = 3.973958333 Factor a perfect square on the left side: (c + 0.375)(c + 0.375) = 3.973958333 Calculate the square root of the right side: 1.993478952 Break this problem into two subproblems by setting (c + 0.375) equal to 1.993478952 and -1.993478952.Subproblem 1
c + 0.375 = 1.993478952 Simplifying c + 0.375 = 1.993478952 Reorder the terms: 0.375 + c = 1.993478952 Solving 0.375 + c = 1.993478952 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + c = 1.993478952 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + c = 1.993478952 + -0.375 c = 1.993478952 + -0.375 Combine like terms: 1.993478952 + -0.375 = 1.618478952 c = 1.618478952 Simplifying c = 1.618478952Subproblem 2
c + 0.375 = -1.993478952 Simplifying c + 0.375 = -1.993478952 Reorder the terms: 0.375 + c = -1.993478952 Solving 0.375 + c = -1.993478952 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + c = -1.993478952 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + c = -1.993478952 + -0.375 c = -1.993478952 + -0.375 Combine like terms: -1.993478952 + -0.375 = -2.368478952 c = -2.368478952 Simplifying c = -2.368478952Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.618478952, -2.368478952}Solution
c = {1.618478952, -2.368478952}
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