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Simplifying 64c2 + 4c + -12 = 0 Reorder the terms: -12 + 4c + 64c2 = 0 Solving -12 + 4c + 64c2 = 0 Solving for variable 'c'. Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + c + 16c2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + c + 16c2)' equal to zero and attempt to solve: Simplifying -3 + c + 16c2 = 0 Solving -3 + c + 16c2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + 0.0625c + c2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + 0.0625c + 0.1875 + c2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + 0.0625c + c2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + 0.0625c + c2 = 0 + 0.1875 0.0625c + c2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 0.0625c + c2 = 0.1875 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. 0.0625c + 0.25 + c2 = 0.1875 + 0.25 Reorder the terms: 0.25 + 0.0625c + c2 = 0.1875 + 0.25 Combine like terms: 0.1875 + 0.25 = 0.4375 0.25 + 0.0625c + c2 = 0.4375 Factor a perfect square on the left side: (c + 0.5)(c + 0.5) = 0.4375 Calculate the square root of the right side: 0.661437828 Break this problem into two subproblems by setting (c + 0.5) equal to 0.661437828 and -0.661437828.Subproblem 1
c + 0.5 = 0.661437828 Simplifying c + 0.5 = 0.661437828 Reorder the terms: 0.5 + c = 0.661437828 Solving 0.5 + c = 0.661437828 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 = 0.661437828 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + c = 0.661437828 + -0.5 c = 0.661437828 + -0.5 Combine like terms: 0.661437828 + -0.5 = 0.161437828 c = 0.161437828 Simplifying c = 0.161437828Subproblem 2
c + 0.5 = -0.661437828 Simplifying c + 0.5 = -0.661437828 Reorder the terms: 0.5 + c = -0.661437828 Solving 0.5 + c = -0.661437828 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 = -0.661437828 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + c = -0.661437828 + -0.5 c = -0.661437828 + -0.5 Combine like terms: -0.661437828 + -0.5 = -1.161437828 c = -1.161437828 Simplifying c = -1.161437828Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.161437828, -1.161437828}Solution
c = {0.161437828, -1.161437828}
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