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Simplifying (12c3 + -5c2 + -2c) + [(7c3 + -2c2 + c) + -1(4c3 + 4c)] = 0 Reorder the terms: (-2c + -5c2 + 12c3) + [(7c3 + -2c2 + c) + -1(4c3 + 4c)] = 0 Remove parenthesis around (-2c + -5c2 + 12c3) -2c + -5c2 + 12c3 + [(7c3 + -2c2 + c) + -1(4c3 + 4c)] = 0 Reorder the terms: -2c + -5c2 + 12c3 + [(c + -2c2 + 7c3) + -1(4c3 + 4c)] = 0 Remove parenthesis around (c + -2c2 + 7c3) -2c + -5c2 + 12c3 + [c + -2c2 + 7c3 + -1(4c3 + 4c)] = 0 Reorder the terms: -2c + -5c2 + 12c3 + [c + -2c2 + 7c3 + -1(4c + 4c3)] = 0 -2c + -5c2 + 12c3 + [c + -2c2 + 7c3 + (4c * -1 + 4c3 * -1)] = 0 -2c + -5c2 + 12c3 + [c + -2c2 + 7c3 + (-4c + -4c3)] = 0 Reorder the terms: -2c + -5c2 + 12c3 + [c + -4c + -2c2 + 7c3 + -4c3] = 0 Combine like terms: c + -4c = -3c -2c + -5c2 + 12c3 + [-3c + -2c2 + 7c3 + -4c3] = 0 Combine like terms: 7c3 + -4c3 = 3c3 -2c + -5c2 + 12c3 + [-3c + -2c2 + 3c3] = 0 Remove brackets around [-3c + -2c2 + 3c3] -2c + -5c2 + 12c3 + -3c + -2c2 + 3c3 = 0 Reorder the terms: -2c + -3c + -5c2 + -2c2 + 12c3 + 3c3 = 0 Combine like terms: -2c + -3c = -5c -5c + -5c2 + -2c2 + 12c3 + 3c3 = 0 Combine like terms: -5c2 + -2c2 = -7c2 -5c + -7c2 + 12c3 + 3c3 = 0 Combine like terms: 12c3 + 3c3 = 15c3 -5c + -7c2 + 15c3 = 0 Solving -5c + -7c2 + 15c3 = 0 Solving for variable 'c'. Factor out the Greatest Common Factor (GCF), 'c'. c(-5 + -7c + 15c2) = 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 '(-5 + -7c + 15c2)' equal to zero and attempt to solve: Simplifying -5 + -7c + 15c2 = 0 Solving -5 + -7c + 15c2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.3333333333 + -0.4666666667c + c2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -0.4666666667c + 0.3333333333 + c2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -0.4666666667c + c2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -0.4666666667c + c2 = 0 + 0.3333333333 -0.4666666667c + c2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -0.4666666667c + c2 = 0.3333333333 The c term is -0.4666666667c. Take half its coefficient (-0.2333333334). Square it (0.05444444448) and add it to both sides. Add '0.05444444448' to each side of the equation. -0.4666666667c + 0.05444444448 + c2 = 0.3333333333 + 0.05444444448 Reorder the terms: 0.05444444448 + -0.4666666667c + c2 = 0.3333333333 + 0.05444444448 Combine like terms: 0.3333333333 + 0.05444444448 = 0.38777777778 0.05444444448 + -0.4666666667c + c2 = 0.38777777778 Factor a perfect square on the left side: (c + -0.2333333334)(c + -0.2333333334) = 0.38777777778 Calculate the square root of the right side: 0.622718056 Break this problem into two subproblems by setting (c + -0.2333333334) equal to 0.622718056 and -0.622718056.Subproblem 1
c + -0.2333333334 = 0.622718056 Simplifying c + -0.2333333334 = 0.622718056 Reorder the terms: -0.2333333334 + c = 0.622718056 Solving -0.2333333334 + c = 0.622718056 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.2333333334' to each side of the equation. -0.2333333334 + 0.2333333334 + c = 0.622718056 + 0.2333333334 Combine like terms: -0.2333333334 + 0.2333333334 = 0.0000000000 0.0000000000 + c = 0.622718056 + 0.2333333334 c = 0.622718056 + 0.2333333334 Combine like terms: 0.622718056 + 0.2333333334 = 0.8560513894 c = 0.8560513894 Simplifying c = 0.8560513894Subproblem 2
c + -0.2333333334 = -0.622718056 Simplifying c + -0.2333333334 = -0.622718056 Reorder the terms: -0.2333333334 + c = -0.622718056 Solving -0.2333333334 + c = -0.622718056 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.2333333334' to each side of the equation. -0.2333333334 + 0.2333333334 + c = -0.622718056 + 0.2333333334 Combine like terms: -0.2333333334 + 0.2333333334 = 0.0000000000 0.0000000000 + c = -0.622718056 + 0.2333333334 c = -0.622718056 + 0.2333333334 Combine like terms: -0.622718056 + 0.2333333334 = -0.3893847226 c = -0.3893847226 Simplifying c = -0.3893847226Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.8560513894, -0.3893847226}Solution
c = {0, 0.8560513894, -0.3893847226}
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