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Simplifying (3c + -4c2 + c3) + -1(5c2 + 8c3 + -6c) = 0 Remove parenthesis around (3c + -4c2 + c3) 3c + -4c2 + c3 + -1(5c2 + 8c3 + -6c) = 0 Reorder the terms: 3c + -4c2 + c3 + -1(-6c + 5c2 + 8c3) = 0 3c + -4c2 + c3 + (-6c * -1 + 5c2 * -1 + 8c3 * -1) = 0 3c + -4c2 + c3 + (6c + -5c2 + -8c3) = 0 Reorder the terms: 3c + 6c + -4c2 + -5c2 + c3 + -8c3 = 0 Combine like terms: 3c + 6c = 9c 9c + -4c2 + -5c2 + c3 + -8c3 = 0 Combine like terms: -4c2 + -5c2 = -9c2 9c + -9c2 + c3 + -8c3 = 0 Combine like terms: c3 + -8c3 = -7c3 9c + -9c2 + -7c3 = 0 Solving 9c + -9c2 + -7c3 = 0 Solving for variable 'c'. Factor out the Greatest Common Factor (GCF), 'c'. c(9 + -9c + -7c2) = 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 '(9 + -9c + -7c2)' equal to zero and attempt to solve: Simplifying 9 + -9c + -7c2 = 0 Solving 9 + -9c + -7c2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1.285714286 + 1.285714286c + c2 = 0 Move the constant term to the right: Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286c + 1.285714286 + c2 = 0 + 1.285714286 Reorder the terms: -1.285714286 + 1.285714286 + 1.285714286c + c2 = 0 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + 1.285714286c + c2 = 0 + 1.285714286 1.285714286c + c2 = 0 + 1.285714286 Combine like terms: 0 + 1.285714286 = 1.285714286 1.285714286c + c2 = 1.285714286 The c term is 1.285714286c. Take half its coefficient (0.642857143). Square it (0.4132653063) and add it to both sides. Add '0.4132653063' to each side of the equation. 1.285714286c + 0.4132653063 + c2 = 1.285714286 + 0.4132653063 Reorder the terms: 0.4132653063 + 1.285714286c + c2 = 1.285714286 + 0.4132653063 Combine like terms: 1.285714286 + 0.4132653063 = 1.6989795923 0.4132653063 + 1.285714286c + c2 = 1.6989795923 Factor a perfect square on the left side: (c + 0.642857143)(c + 0.642857143) = 1.6989795923 Calculate the square root of the right side: 1.303449114 Break this problem into two subproblems by setting (c + 0.642857143) equal to 1.303449114 and -1.303449114.Subproblem 1
c + 0.642857143 = 1.303449114 Simplifying c + 0.642857143 = 1.303449114 Reorder the terms: 0.642857143 + c = 1.303449114 Solving 0.642857143 + c = 1.303449114 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.642857143' to each side of the equation. 0.642857143 + -0.642857143 + c = 1.303449114 + -0.642857143 Combine like terms: 0.642857143 + -0.642857143 = 0.000000000 0.000000000 + c = 1.303449114 + -0.642857143 c = 1.303449114 + -0.642857143 Combine like terms: 1.303449114 + -0.642857143 = 0.660591971 c = 0.660591971 Simplifying c = 0.660591971Subproblem 2
c + 0.642857143 = -1.303449114 Simplifying c + 0.642857143 = -1.303449114 Reorder the terms: 0.642857143 + c = -1.303449114 Solving 0.642857143 + c = -1.303449114 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.642857143' to each side of the equation. 0.642857143 + -0.642857143 + c = -1.303449114 + -0.642857143 Combine like terms: 0.642857143 + -0.642857143 = 0.000000000 0.000000000 + c = -1.303449114 + -0.642857143 c = -1.303449114 + -0.642857143 Combine like terms: -1.303449114 + -0.642857143 = -1.946306257 c = -1.946306257 Simplifying c = -1.946306257Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.660591971, -1.946306257}Solution
c = {0, 0.660591971, -1.946306257}
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