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Simplifying 84c + 168 = (4c + 24) * 1c Reorder the terms: 168 + 84c = (4c + 24) * 1c Reorder the terms: 168 + 84c = (24 + 4c) * 1c Reorder the terms for easier multiplication: 168 + 84c = 1c(24 + 4c) 168 + 84c = (24 * 1c + 4c * 1c) 168 + 84c = (24c + 4c2) Solving 168 + 84c = 24c + 4c2 Solving for variable 'c'. Combine like terms: 84c + -24c = 60c 168 + 60c + -4c2 = 24c + 4c2 + -24c + -4c2 Reorder the terms: 168 + 60c + -4c2 = 24c + -24c + 4c2 + -4c2 Combine like terms: 24c + -24c = 0 168 + 60c + -4c2 = 0 + 4c2 + -4c2 168 + 60c + -4c2 = 4c2 + -4c2 Combine like terms: 4c2 + -4c2 = 0 168 + 60c + -4c2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(42 + 15c + -1c2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(42 + 15c + -1c2)' equal to zero and attempt to solve: Simplifying 42 + 15c + -1c2 = 0 Solving 42 + 15c + -1c2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -42 + -15c + c2 = 0 Move the constant term to the right: Add '42' to each side of the equation. -42 + -15c + 42 + c2 = 0 + 42 Reorder the terms: -42 + 42 + -15c + c2 = 0 + 42 Combine like terms: -42 + 42 = 0 0 + -15c + c2 = 0 + 42 -15c + c2 = 0 + 42 Combine like terms: 0 + 42 = 42 -15c + c2 = 42 The c term is -15c. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15c + 56.25 + c2 = 42 + 56.25 Reorder the terms: 56.25 + -15c + c2 = 42 + 56.25 Combine like terms: 42 + 56.25 = 98.25 56.25 + -15c + c2 = 98.25 Factor a perfect square on the left side: (c + -7.5)(c + -7.5) = 98.25 Calculate the square root of the right side: 9.912113801 Break this problem into two subproblems by setting (c + -7.5) equal to 9.912113801 and -9.912113801.Subproblem 1
c + -7.5 = 9.912113801 Simplifying c + -7.5 = 9.912113801 Reorder the terms: -7.5 + c = 9.912113801 Solving -7.5 + c = 9.912113801 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + c = 9.912113801 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + c = 9.912113801 + 7.5 c = 9.912113801 + 7.5 Combine like terms: 9.912113801 + 7.5 = 17.412113801 c = 17.412113801 Simplifying c = 17.412113801Subproblem 2
c + -7.5 = -9.912113801 Simplifying c + -7.5 = -9.912113801 Reorder the terms: -7.5 + c = -9.912113801 Solving -7.5 + c = -9.912113801 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + c = -9.912113801 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + c = -9.912113801 + 7.5 c = -9.912113801 + 7.5 Combine like terms: -9.912113801 + 7.5 = -2.412113801 c = -2.412113801 Simplifying c = -2.412113801Solution
The solution to the problem is based on the solutions from the subproblems. c = {17.412113801, -2.412113801}Solution
c = {17.412113801, -2.412113801}
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