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Simplifying 3c2 + 11c + -10 + 14c2 + -7c + 9 + c + -3 = 0 Reorder the terms: -10 + 9 + -3 + 11c + -7c + c + 3c2 + 14c2 = 0 Combine like terms: -10 + 9 = -1 -1 + -3 + 11c + -7c + c + 3c2 + 14c2 = 0 Combine like terms: -1 + -3 = -4 -4 + 11c + -7c + c + 3c2 + 14c2 = 0 Combine like terms: 11c + -7c = 4c -4 + 4c + c + 3c2 + 14c2 = 0 Combine like terms: 4c + c = 5c -4 + 5c + 3c2 + 14c2 = 0 Combine like terms: 3c2 + 14c2 = 17c2 -4 + 5c + 17c2 = 0 Solving -4 + 5c + 17c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 17 the coefficient of the squared term: Divide each side by '17'. -0.2352941176 + 0.2941176471c + c2 = 0 Move the constant term to the right: Add '0.2352941176' to each side of the equation. -0.2352941176 + 0.2941176471c + 0.2352941176 + c2 = 0 + 0.2352941176 Reorder the terms: -0.2352941176 + 0.2352941176 + 0.2941176471c + c2 = 0 + 0.2352941176 Combine like terms: -0.2352941176 + 0.2352941176 = 0.0000000000 0.0000000000 + 0.2941176471c + c2 = 0 + 0.2352941176 0.2941176471c + c2 = 0 + 0.2352941176 Combine like terms: 0 + 0.2352941176 = 0.2352941176 0.2941176471c + c2 = 0.2352941176 The c term is 0.2941176471c. Take half its coefficient (0.1470588236). Square it (0.02162629760) and add it to both sides. Add '0.02162629760' to each side of the equation. 0.2941176471c + 0.02162629760 + c2 = 0.2352941176 + 0.02162629760 Reorder the terms: 0.02162629760 + 0.2941176471c + c2 = 0.2352941176 + 0.02162629760 Combine like terms: 0.2352941176 + 0.02162629760 = 0.2569204152 0.02162629760 + 0.2941176471c + c2 = 0.2569204152 Factor a perfect square on the left side: (c + 0.1470588236)(c + 0.1470588236) = 0.2569204152 Calculate the square root of the right side: 0.506873175 Break this problem into two subproblems by setting (c + 0.1470588236) equal to 0.506873175 and -0.506873175.Subproblem 1
c + 0.1470588236 = 0.506873175 Simplifying c + 0.1470588236 = 0.506873175 Reorder the terms: 0.1470588236 + c = 0.506873175 Solving 0.1470588236 + c = 0.506873175 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.1470588236' to each side of the equation. 0.1470588236 + -0.1470588236 + c = 0.506873175 + -0.1470588236 Combine like terms: 0.1470588236 + -0.1470588236 = 0.0000000000 0.0000000000 + c = 0.506873175 + -0.1470588236 c = 0.506873175 + -0.1470588236 Combine like terms: 0.506873175 + -0.1470588236 = 0.3598143514 c = 0.3598143514 Simplifying c = 0.3598143514Subproblem 2
c + 0.1470588236 = -0.506873175 Simplifying c + 0.1470588236 = -0.506873175 Reorder the terms: 0.1470588236 + c = -0.506873175 Solving 0.1470588236 + c = -0.506873175 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.1470588236' to each side of the equation. 0.1470588236 + -0.1470588236 + c = -0.506873175 + -0.1470588236 Combine like terms: 0.1470588236 + -0.1470588236 = 0.0000000000 0.0000000000 + c = -0.506873175 + -0.1470588236 c = -0.506873175 + -0.1470588236 Combine like terms: -0.506873175 + -0.1470588236 = -0.6539319986 c = -0.6539319986 Simplifying c = -0.6539319986Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.3598143514, -0.6539319986}
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