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Simplifying (3c + 4)(2c + -11) + -106c = (12 + 10c)(12 + -10c) + 12 Reorder the terms: (4 + 3c)(2c + -11) + -106c = (12 + 10c)(12 + -10c) + 12 Reorder the terms: (4 + 3c)(-11 + 2c) + -106c = (12 + 10c)(12 + -10c) + 12 Multiply (4 + 3c) * (-11 + 2c) (4(-11 + 2c) + 3c * (-11 + 2c)) + -106c = (12 + 10c)(12 + -10c) + 12 ((-11 * 4 + 2c * 4) + 3c * (-11 + 2c)) + -106c = (12 + 10c)(12 + -10c) + 12 ((-44 + 8c) + 3c * (-11 + 2c)) + -106c = (12 + 10c)(12 + -10c) + 12 (-44 + 8c + (-11 * 3c + 2c * 3c)) + -106c = (12 + 10c)(12 + -10c) + 12 (-44 + 8c + (-33c + 6c2)) + -106c = (12 + 10c)(12 + -10c) + 12 Combine like terms: 8c + -33c = -25c (-44 + -25c + 6c2) + -106c = (12 + 10c)(12 + -10c) + 12 Reorder the terms: -44 + -25c + -106c + 6c2 = (12 + 10c)(12 + -10c) + 12 Combine like terms: -25c + -106c = -131c -44 + -131c + 6c2 = (12 + 10c)(12 + -10c) + 12 Multiply (12 + 10c) * (12 + -10c) -44 + -131c + 6c2 = (12(12 + -10c) + 10c * (12 + -10c)) + 12 -44 + -131c + 6c2 = ((12 * 12 + -10c * 12) + 10c * (12 + -10c)) + 12 -44 + -131c + 6c2 = ((144 + -120c) + 10c * (12 + -10c)) + 12 -44 + -131c + 6c2 = (144 + -120c + (12 * 10c + -10c * 10c)) + 12 -44 + -131c + 6c2 = (144 + -120c + (120c + -100c2)) + 12 Combine like terms: -120c + 120c = 0 -44 + -131c + 6c2 = (144 + 0 + -100c2) + 12 -44 + -131c + 6c2 = (144 + -100c2) + 12 Reorder the terms: -44 + -131c + 6c2 = 144 + 12 + -100c2 Combine like terms: 144 + 12 = 156 -44 + -131c + 6c2 = 156 + -100c2 Solving -44 + -131c + 6c2 = 156 + -100c2 Solving for variable 'c'. Reorder the terms: -44 + -156 + -131c + 6c2 + 100c2 = 156 + -100c2 + -156 + 100c2 Combine like terms: -44 + -156 = -200 -200 + -131c + 6c2 + 100c2 = 156 + -100c2 + -156 + 100c2 Combine like terms: 6c2 + 100c2 = 106c2 -200 + -131c + 106c2 = 156 + -100c2 + -156 + 100c2 Reorder the terms: -200 + -131c + 106c2 = 156 + -156 + -100c2 + 100c2 Combine like terms: 156 + -156 = 0 -200 + -131c + 106c2 = 0 + -100c2 + 100c2 -200 + -131c + 106c2 = -100c2 + 100c2 Combine like terms: -100c2 + 100c2 = 0 -200 + -131c + 106c2 = 0 Begin completing the square. Divide all terms by 106 the coefficient of the squared term: Divide each side by '106'. -1.886792453 + -1.235849057c + c2 = 0 Move the constant term to the right: Add '1.886792453' to each side of the equation. -1.886792453 + -1.235849057c + 1.886792453 + c2 = 0 + 1.886792453 Reorder the terms: -1.886792453 + 1.886792453 + -1.235849057c + c2 = 0 + 1.886792453 Combine like terms: -1.886792453 + 1.886792453 = 0.000000000 0.000000000 + -1.235849057c + c2 = 0 + 1.886792453 -1.235849057c + c2 = 0 + 1.886792453 Combine like terms: 0 + 1.886792453 = 1.886792453 -1.235849057c + c2 = 1.886792453 The c term is -1.235849057c. Take half its coefficient (-0.6179245285). Square it (0.3818307229) and add it to both sides. Add '0.3818307229' to each side of the equation. -1.235849057c + 0.3818307229 + c2 = 1.886792453 + 0.3818307229 Reorder the terms: 0.3818307229 + -1.235849057c + c2 = 1.886792453 + 0.3818307229 Combine like terms: 1.886792453 + 0.3818307229 = 2.2686231759 0.3818307229 + -1.235849057c + c2 = 2.2686231759 Factor a perfect square on the left side: (c + -0.6179245285)(c + -0.6179245285) = 2.2686231759 Calculate the square root of the right side: 1.506194933 Break this problem into two subproblems by setting (c + -0.6179245285) equal to 1.506194933 and -1.506194933.Subproblem 1
c + -0.6179245285 = 1.506194933 Simplifying c + -0.6179245285 = 1.506194933 Reorder the terms: -0.6179245285 + c = 1.506194933 Solving -0.6179245285 + c = 1.506194933 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.6179245285' to each side of the equation. -0.6179245285 + 0.6179245285 + c = 1.506194933 + 0.6179245285 Combine like terms: -0.6179245285 + 0.6179245285 = 0.0000000000 0.0000000000 + c = 1.506194933 + 0.6179245285 c = 1.506194933 + 0.6179245285 Combine like terms: 1.506194933 + 0.6179245285 = 2.1241194615 c = 2.1241194615 Simplifying c = 2.1241194615Subproblem 2
c + -0.6179245285 = -1.506194933 Simplifying c + -0.6179245285 = -1.506194933 Reorder the terms: -0.6179245285 + c = -1.506194933 Solving -0.6179245285 + c = -1.506194933 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.6179245285' to each side of the equation. -0.6179245285 + 0.6179245285 + c = -1.506194933 + 0.6179245285 Combine like terms: -0.6179245285 + 0.6179245285 = 0.0000000000 0.0000000000 + c = -1.506194933 + 0.6179245285 c = -1.506194933 + 0.6179245285 Combine like terms: -1.506194933 + 0.6179245285 = -0.8882704045 c = -0.8882704045 Simplifying c = -0.8882704045Solution
The solution to the problem is based on the solutions from the subproblems. c = {2.1241194615, -0.8882704045}
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