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Simplifying 6c3 + 2c2 + -192c = 0 Reorder the terms: -192c + 2c2 + 6c3 = 0 Solving -192c + 2c2 + 6c3 = 0 Solving for variable 'c'. Factor out the Greatest Common Factor (GCF), '2c'. 2c(-96 + c + 3c2) = 0 Ignore the factor 2.Subproblem 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 '(-96 + c + 3c2)' equal to zero and attempt to solve: Simplifying -96 + c + 3c2 = 0 Solving -96 + c + 3c2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -32 + 0.3333333333c + c2 = 0 Move the constant term to the right: Add '32' to each side of the equation. -32 + 0.3333333333c + 32 + c2 = 0 + 32 Reorder the terms: -32 + 32 + 0.3333333333c + c2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 0.3333333333c + c2 = 0 + 32 0.3333333333c + c2 = 0 + 32 Combine like terms: 0 + 32 = 32 0.3333333333c + c2 = 32 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. 0.3333333333c + 0.25 + c2 = 32 + 0.25 Reorder the terms: 0.25 + 0.3333333333c + c2 = 32 + 0.25 Combine like terms: 32 + 0.25 = 32.25 0.25 + 0.3333333333c + c2 = 32.25 Factor a perfect square on the left side: (c + 0.5)(c + 0.5) = 32.25 Calculate the square root of the right side: 5.678908346 Break this problem into two subproblems by setting (c + 0.5) equal to 5.678908346 and -5.678908346.Subproblem 1
c + 0.5 = 5.678908346 Simplifying c + 0.5 = 5.678908346 Reorder the terms: 0.5 + c = 5.678908346 Solving 0.5 + c = 5.678908346 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 = 5.678908346 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + c = 5.678908346 + -0.5 c = 5.678908346 + -0.5 Combine like terms: 5.678908346 + -0.5 = 5.178908346 c = 5.178908346 Simplifying c = 5.178908346Subproblem 2
c + 0.5 = -5.678908346 Simplifying c + 0.5 = -5.678908346 Reorder the terms: 0.5 + c = -5.678908346 Solving 0.5 + c = -5.678908346 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 = -5.678908346 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + c = -5.678908346 + -0.5 c = -5.678908346 + -0.5 Combine like terms: -5.678908346 + -0.5 = -6.178908346 c = -6.178908346 Simplifying c = -6.178908346Solution
The solution to the problem is based on the solutions from the subproblems. c = {5.178908346, -6.178908346}Solution
c = {0, 5.178908346, -6.178908346}
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