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Simplifying 34 + -1(3c + 4) = 2(c + 6) * c Reorder the terms: 34 + -1(4 + 3c) = 2(c + 6) * c 34 + (4 * -1 + 3c * -1) = 2(c + 6) * c 34 + (-4 + -3c) = 2(c + 6) * c Combine like terms: 34 + -4 = 30 30 + -3c = 2(c + 6) * c Reorder the terms: 30 + -3c = 2(6 + c) * c Reorder the terms for easier multiplication: 30 + -3c = 2c(6 + c) 30 + -3c = (6 * 2c + c * 2c) 30 + -3c = (12c + 2c2) Solving 30 + -3c = 12c + 2c2 Solving for variable 'c'. Combine like terms: -3c + -12c = -15c 30 + -15c + -2c2 = 12c + 2c2 + -12c + -2c2 Reorder the terms: 30 + -15c + -2c2 = 12c + -12c + 2c2 + -2c2 Combine like terms: 12c + -12c = 0 30 + -15c + -2c2 = 0 + 2c2 + -2c2 30 + -15c + -2c2 = 2c2 + -2c2 Combine like terms: 2c2 + -2c2 = 0 30 + -15c + -2c2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -15 + 7.5c + c2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 7.5c + 15 + c2 = 0 + 15 Reorder the terms: -15 + 15 + 7.5c + c2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 7.5c + c2 = 0 + 15 7.5c + c2 = 0 + 15 Combine like terms: 0 + 15 = 15 7.5c + c2 = 15 The c term is 7.5c. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5c + 14.0625 + c2 = 15 + 14.0625 Reorder the terms: 14.0625 + 7.5c + c2 = 15 + 14.0625 Combine like terms: 15 + 14.0625 = 29.0625 14.0625 + 7.5c + c2 = 29.0625 Factor a perfect square on the left side: (c + 3.75)(c + 3.75) = 29.0625 Calculate the square root of the right side: 5.390964663 Break this problem into two subproblems by setting (c + 3.75) equal to 5.390964663 and -5.390964663.Subproblem 1
c + 3.75 = 5.390964663 Simplifying c + 3.75 = 5.390964663 Reorder the terms: 3.75 + c = 5.390964663 Solving 3.75 + c = 5.390964663 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + c = 5.390964663 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + c = 5.390964663 + -3.75 c = 5.390964663 + -3.75 Combine like terms: 5.390964663 + -3.75 = 1.640964663 c = 1.640964663 Simplifying c = 1.640964663Subproblem 2
c + 3.75 = -5.390964663 Simplifying c + 3.75 = -5.390964663 Reorder the terms: 3.75 + c = -5.390964663 Solving 3.75 + c = -5.390964663 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + c = -5.390964663 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + c = -5.390964663 + -3.75 c = -5.390964663 + -3.75 Combine like terms: -5.390964663 + -3.75 = -9.140964663 c = -9.140964663 Simplifying c = -9.140964663Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.640964663, -9.140964663}
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