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Simplifying 15k2 + -7k + -3 + -8 + -2k2 + -4k = 0 Reorder the terms: -3 + -8 + -7k + -4k + 15k2 + -2k2 = 0 Combine like terms: -3 + -8 = -11 -11 + -7k + -4k + 15k2 + -2k2 = 0 Combine like terms: -7k + -4k = -11k -11 + -11k + 15k2 + -2k2 = 0 Combine like terms: 15k2 + -2k2 = 13k2 -11 + -11k + 13k2 = 0 Solving -11 + -11k + 13k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -0.8461538462 + -0.8461538462k + k2 = 0 Move the constant term to the right: Add '0.8461538462' to each side of the equation. -0.8461538462 + -0.8461538462k + 0.8461538462 + k2 = 0 + 0.8461538462 Reorder the terms: -0.8461538462 + 0.8461538462 + -0.8461538462k + k2 = 0 + 0.8461538462 Combine like terms: -0.8461538462 + 0.8461538462 = 0.0000000000 0.0000000000 + -0.8461538462k + k2 = 0 + 0.8461538462 -0.8461538462k + k2 = 0 + 0.8461538462 Combine like terms: 0 + 0.8461538462 = 0.8461538462 -0.8461538462k + k2 = 0.8461538462 The k term is -0.8461538462k. Take half its coefficient (-0.4230769231). Square it (0.1789940829) and add it to both sides. Add '0.1789940829' to each side of the equation. -0.8461538462k + 0.1789940829 + k2 = 0.8461538462 + 0.1789940829 Reorder the terms: 0.1789940829 + -0.8461538462k + k2 = 0.8461538462 + 0.1789940829 Combine like terms: 0.8461538462 + 0.1789940829 = 1.0251479291 0.1789940829 + -0.8461538462k + k2 = 1.0251479291 Factor a perfect square on the left side: (k + -0.4230769231)(k + -0.4230769231) = 1.0251479291 Calculate the square root of the right side: 1.012495891 Break this problem into two subproblems by setting (k + -0.4230769231) equal to 1.012495891 and -1.012495891.Subproblem 1
k + -0.4230769231 = 1.012495891 Simplifying k + -0.4230769231 = 1.012495891 Reorder the terms: -0.4230769231 + k = 1.012495891 Solving -0.4230769231 + k = 1.012495891 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.4230769231' to each side of the equation. -0.4230769231 + 0.4230769231 + k = 1.012495891 + 0.4230769231 Combine like terms: -0.4230769231 + 0.4230769231 = 0.0000000000 0.0000000000 + k = 1.012495891 + 0.4230769231 k = 1.012495891 + 0.4230769231 Combine like terms: 1.012495891 + 0.4230769231 = 1.4355728141 k = 1.4355728141 Simplifying k = 1.4355728141Subproblem 2
k + -0.4230769231 = -1.012495891 Simplifying k + -0.4230769231 = -1.012495891 Reorder the terms: -0.4230769231 + k = -1.012495891 Solving -0.4230769231 + k = -1.012495891 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.4230769231' to each side of the equation. -0.4230769231 + 0.4230769231 + k = -1.012495891 + 0.4230769231 Combine like terms: -0.4230769231 + 0.4230769231 = 0.0000000000 0.0000000000 + k = -1.012495891 + 0.4230769231 k = -1.012495891 + 0.4230769231 Combine like terms: -1.012495891 + 0.4230769231 = -0.5894189679 k = -0.5894189679 Simplifying k = -0.5894189679Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.4355728141, -0.5894189679}
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