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Simplifying (41.5 + c)(10c + 1) = 180 Reorder the terms: (41.5 + c)(1 + 10c) = 180 Multiply (41.5 + c) * (1 + 10c) (41.5(1 + 10c) + c(1 + 10c)) = 180 ((1 * 41.5 + 10c * 41.5) + c(1 + 10c)) = 180 ((41.5 + 415c) + c(1 + 10c)) = 180 (41.5 + 415c + (1 * c + 10c * c)) = 180 (41.5 + 415c + (1c + 10c2)) = 180 Combine like terms: 415c + 1c = 416c (41.5 + 416c + 10c2) = 180 Solving 41.5 + 416c + 10c2 = 180 Solving for variable 'c'. Reorder the terms: 41.5 + -180 + 416c + 10c2 = 180 + -180 Combine like terms: 41.5 + -180 = -138.5 -138.5 + 416c + 10c2 = 180 + -180 Combine like terms: 180 + -180 = 0 -138.5 + 416c + 10c2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -13.85 + 41.6c + c2 = 0 Move the constant term to the right: Add '13.85' to each side of the equation. -13.85 + 41.6c + 13.85 + c2 = 0 + 13.85 Reorder the terms: -13.85 + 13.85 + 41.6c + c2 = 0 + 13.85 Combine like terms: -13.85 + 13.85 = 0.00 0.00 + 41.6c + c2 = 0 + 13.85 41.6c + c2 = 0 + 13.85 Combine like terms: 0 + 13.85 = 13.85 41.6c + c2 = 13.85 The c term is 41.6c. Take half its coefficient (20.8). Square it (432.64) and add it to both sides. Add '432.64' to each side of the equation. 41.6c + 432.64 + c2 = 13.85 + 432.64 Reorder the terms: 432.64 + 41.6c + c2 = 13.85 + 432.64 Combine like terms: 13.85 + 432.64 = 446.49 432.64 + 41.6c + c2 = 446.49 Factor a perfect square on the left side: (c + 20.8)(c + 20.8) = 446.49 Calculate the square root of the right side: 21.130309984 Break this problem into two subproblems by setting (c + 20.8) equal to 21.130309984 and -21.130309984.Subproblem 1
c + 20.8 = 21.130309984 Simplifying c + 20.8 = 21.130309984 Reorder the terms: 20.8 + c = 21.130309984 Solving 20.8 + c = 21.130309984 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-20.8' to each side of the equation. 20.8 + -20.8 + c = 21.130309984 + -20.8 Combine like terms: 20.8 + -20.8 = 0.0 0.0 + c = 21.130309984 + -20.8 c = 21.130309984 + -20.8 Combine like terms: 21.130309984 + -20.8 = 0.330309984 c = 0.330309984 Simplifying c = 0.330309984Subproblem 2
c + 20.8 = -21.130309984 Simplifying c + 20.8 = -21.130309984 Reorder the terms: 20.8 + c = -21.130309984 Solving 20.8 + c = -21.130309984 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-20.8' to each side of the equation. 20.8 + -20.8 + c = -21.130309984 + -20.8 Combine like terms: 20.8 + -20.8 = 0.0 0.0 + c = -21.130309984 + -20.8 c = -21.130309984 + -20.8 Combine like terms: -21.130309984 + -20.8 = -41.930309984 c = -41.930309984 Simplifying c = -41.930309984Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.330309984, -41.930309984}
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