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Simplifying 18c2 + 21c + 4 = -2c Reorder the terms: 4 + 21c + 18c2 = -2c Solving 4 + 21c + 18c2 = -2c Solving for variable 'c'. Reorder the terms: 4 + 21c + 2c + 18c2 = -2c + 2c Combine like terms: 21c + 2c = 23c 4 + 23c + 18c2 = -2c + 2c Combine like terms: -2c + 2c = 0 4 + 23c + 18c2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.2222222222 + 1.277777778c + c2 = 0 Move the constant term to the right: Add '-0.2222222222' to each side of the equation. 0.2222222222 + 1.277777778c + -0.2222222222 + c2 = 0 + -0.2222222222 Reorder the terms: 0.2222222222 + -0.2222222222 + 1.277777778c + c2 = 0 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + 1.277777778c + c2 = 0 + -0.2222222222 1.277777778c + c2 = 0 + -0.2222222222 Combine like terms: 0 + -0.2222222222 = -0.2222222222 1.277777778c + c2 = -0.2222222222 The c term is 1.277777778c. Take half its coefficient (0.638888889). Square it (0.4081790125) and add it to both sides. Add '0.4081790125' to each side of the equation. 1.277777778c + 0.4081790125 + c2 = -0.2222222222 + 0.4081790125 Reorder the terms: 0.4081790125 + 1.277777778c + c2 = -0.2222222222 + 0.4081790125 Combine like terms: -0.2222222222 + 0.4081790125 = 0.1859567903 0.4081790125 + 1.277777778c + c2 = 0.1859567903 Factor a perfect square on the left side: (c + 0.638888889)(c + 0.638888889) = 0.1859567903 Calculate the square root of the right side: 0.431227075 Break this problem into two subproblems by setting (c + 0.638888889) equal to 0.431227075 and -0.431227075.Subproblem 1
c + 0.638888889 = 0.431227075 Simplifying c + 0.638888889 = 0.431227075 Reorder the terms: 0.638888889 + c = 0.431227075 Solving 0.638888889 + c = 0.431227075 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.638888889' to each side of the equation. 0.638888889 + -0.638888889 + c = 0.431227075 + -0.638888889 Combine like terms: 0.638888889 + -0.638888889 = 0.000000000 0.000000000 + c = 0.431227075 + -0.638888889 c = 0.431227075 + -0.638888889 Combine like terms: 0.431227075 + -0.638888889 = -0.207661814 c = -0.207661814 Simplifying c = -0.207661814Subproblem 2
c + 0.638888889 = -0.431227075 Simplifying c + 0.638888889 = -0.431227075 Reorder the terms: 0.638888889 + c = -0.431227075 Solving 0.638888889 + c = -0.431227075 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.638888889' to each side of the equation. 0.638888889 + -0.638888889 + c = -0.431227075 + -0.638888889 Combine like terms: 0.638888889 + -0.638888889 = 0.000000000 0.000000000 + c = -0.431227075 + -0.638888889 c = -0.431227075 + -0.638888889 Combine like terms: -0.431227075 + -0.638888889 = -1.070115964 c = -1.070115964 Simplifying c = -1.070115964Solution
The solution to the problem is based on the solutions from the subproblems. c = {-0.207661814, -1.070115964}
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