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Simplifying 9c + c(2c + -3) = 9 Reorder the terms: 9c + c(-3 + 2c) = 9 9c + (-3 * c + 2c * c) = 9 9c + (-3c + 2c2) = 9 Combine like terms: 9c + -3c = 6c 6c + 2c2 = 9 Solving 6c + 2c2 = 9 Solving for variable 'c'. Reorder the terms: -9 + 6c + 2c2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + 6c + 2c2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + 3c + c2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 3c + 4.5 + c2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 3c + c2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 3c + c2 = 0 + 4.5 3c + c2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 3c + c2 = 4.5 The c term is 3c. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3c + 2.25 + c2 = 4.5 + 2.25 Reorder the terms: 2.25 + 3c + c2 = 4.5 + 2.25 Combine like terms: 4.5 + 2.25 = 6.75 2.25 + 3c + c2 = 6.75 Factor a perfect square on the left side: (c + 1.5)(c + 1.5) = 6.75 Calculate the square root of the right side: 2.598076211 Break this problem into two subproblems by setting (c + 1.5) equal to 2.598076211 and -2.598076211.Subproblem 1
c + 1.5 = 2.598076211 Simplifying c + 1.5 = 2.598076211 Reorder the terms: 1.5 + c = 2.598076211 Solving 1.5 + c = 2.598076211 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + c = 2.598076211 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + c = 2.598076211 + -1.5 c = 2.598076211 + -1.5 Combine like terms: 2.598076211 + -1.5 = 1.098076211 c = 1.098076211 Simplifying c = 1.098076211Subproblem 2
c + 1.5 = -2.598076211 Simplifying c + 1.5 = -2.598076211 Reorder the terms: 1.5 + c = -2.598076211 Solving 1.5 + c = -2.598076211 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + c = -2.598076211 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + c = -2.598076211 + -1.5 c = -2.598076211 + -1.5 Combine like terms: -2.598076211 + -1.5 = -4.098076211 c = -4.098076211 Simplifying c = -4.098076211Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.098076211, -4.098076211}
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