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Simplifying 53 + -1(2c + 4) = 4(c + 7) * c Reorder the terms: 53 + -1(4 + 2c) = 4(c + 7) * c 53 + (4 * -1 + 2c * -1) = 4(c + 7) * c 53 + (-4 + -2c) = 4(c + 7) * c Combine like terms: 53 + -4 = 49 49 + -2c = 4(c + 7) * c Reorder the terms: 49 + -2c = 4(7 + c) * c Reorder the terms for easier multiplication: 49 + -2c = 4c(7 + c) 49 + -2c = (7 * 4c + c * 4c) 49 + -2c = (28c + 4c2) Solving 49 + -2c = 28c + 4c2 Solving for variable 'c'. Combine like terms: -2c + -28c = -30c 49 + -30c + -4c2 = 28c + 4c2 + -28c + -4c2 Reorder the terms: 49 + -30c + -4c2 = 28c + -28c + 4c2 + -4c2 Combine like terms: 28c + -28c = 0 49 + -30c + -4c2 = 0 + 4c2 + -4c2 49 + -30c + -4c2 = 4c2 + -4c2 Combine like terms: 4c2 + -4c2 = 0 49 + -30c + -4c2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -12.25 + 7.5c + c2 = 0 Move the constant term to the right: Add '12.25' to each side of the equation. -12.25 + 7.5c + 12.25 + c2 = 0 + 12.25 Reorder the terms: -12.25 + 12.25 + 7.5c + c2 = 0 + 12.25 Combine like terms: -12.25 + 12.25 = 0.00 0.00 + 7.5c + c2 = 0 + 12.25 7.5c + c2 = 0 + 12.25 Combine like terms: 0 + 12.25 = 12.25 7.5c + c2 = 12.25 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 = 12.25 + 14.0625 Reorder the terms: 14.0625 + 7.5c + c2 = 12.25 + 14.0625 Combine like terms: 12.25 + 14.0625 = 26.3125 14.0625 + 7.5c + c2 = 26.3125 Factor a perfect square on the left side: (c + 3.75)(c + 3.75) = 26.3125 Calculate the square root of the right side: 5.129571132 Break this problem into two subproblems by setting (c + 3.75) equal to 5.129571132 and -5.129571132.Subproblem 1
c + 3.75 = 5.129571132 Simplifying c + 3.75 = 5.129571132 Reorder the terms: 3.75 + c = 5.129571132 Solving 3.75 + c = 5.129571132 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 = 5.129571132 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + c = 5.129571132 + -3.75 c = 5.129571132 + -3.75 Combine like terms: 5.129571132 + -3.75 = 1.379571132 c = 1.379571132 Simplifying c = 1.379571132Subproblem 2
c + 3.75 = -5.129571132 Simplifying c + 3.75 = -5.129571132 Reorder the terms: 3.75 + c = -5.129571132 Solving 3.75 + c = -5.129571132 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 = -5.129571132 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + c = -5.129571132 + -3.75 c = -5.129571132 + -3.75 Combine like terms: -5.129571132 + -3.75 = -8.879571132 c = -8.879571132 Simplifying c = -8.879571132Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.379571132, -8.879571132}
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