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Simplifying c(c + 12) = 2c + 18 Reorder the terms: c(12 + c) = 2c + 18 (12 * c + c * c) = 2c + 18 (12c + c2) = 2c + 18 Reorder the terms: 12c + c2 = 18 + 2c Solving 12c + c2 = 18 + 2c Solving for variable 'c'. Reorder the terms: -18 + 12c + -2c + c2 = 18 + 2c + -18 + -2c Combine like terms: 12c + -2c = 10c -18 + 10c + c2 = 18 + 2c + -18 + -2c Reorder the terms: -18 + 10c + c2 = 18 + -18 + 2c + -2c Combine like terms: 18 + -18 = 0 -18 + 10c + c2 = 0 + 2c + -2c -18 + 10c + c2 = 2c + -2c Combine like terms: 2c + -2c = 0 -18 + 10c + c2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 10c + 18 + c2 = 0 + 18 Reorder the terms: -18 + 18 + 10c + c2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 10c + c2 = 0 + 18 10c + c2 = 0 + 18 Combine like terms: 0 + 18 = 18 10c + c2 = 18 The c term is 10c. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10c + 25 + c2 = 18 + 25 Reorder the terms: 25 + 10c + c2 = 18 + 25 Combine like terms: 18 + 25 = 43 25 + 10c + c2 = 43 Factor a perfect square on the left side: (c + 5)(c + 5) = 43 Calculate the square root of the right side: 6.557438524 Break this problem into two subproblems by setting (c + 5) equal to 6.557438524 and -6.557438524.Subproblem 1
c + 5 = 6.557438524 Simplifying c + 5 = 6.557438524 Reorder the terms: 5 + c = 6.557438524 Solving 5 + c = 6.557438524 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + c = 6.557438524 + -5 Combine like terms: 5 + -5 = 0 0 + c = 6.557438524 + -5 c = 6.557438524 + -5 Combine like terms: 6.557438524 + -5 = 1.557438524 c = 1.557438524 Simplifying c = 1.557438524Subproblem 2
c + 5 = -6.557438524 Simplifying c + 5 = -6.557438524 Reorder the terms: 5 + c = -6.557438524 Solving 5 + c = -6.557438524 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + c = -6.557438524 + -5 Combine like terms: 5 + -5 = 0 0 + c = -6.557438524 + -5 c = -6.557438524 + -5 Combine like terms: -6.557438524 + -5 = -11.557438524 c = -11.557438524 Simplifying c = -11.557438524Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.557438524, -11.557438524}
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