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Simplifying 13k2 + 2 + -2k + -3 = 0 Reorder the terms: 2 + -3 + -2k + 13k2 = 0 Combine like terms: 2 + -3 = -1 -1 + -2k + 13k2 = 0 Solving -1 + -2k + 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'. -0.07692307692 + -0.1538461538k + k2 = 0 Move the constant term to the right: Add '0.07692307692' to each side of the equation. -0.07692307692 + -0.1538461538k + 0.07692307692 + k2 = 0 + 0.07692307692 Reorder the terms: -0.07692307692 + 0.07692307692 + -0.1538461538k + k2 = 0 + 0.07692307692 Combine like terms: -0.07692307692 + 0.07692307692 = 0.00000000000 0.00000000000 + -0.1538461538k + k2 = 0 + 0.07692307692 -0.1538461538k + k2 = 0 + 0.07692307692 Combine like terms: 0 + 0.07692307692 = 0.07692307692 -0.1538461538k + k2 = 0.07692307692 The k term is -0.1538461538k. Take half its coefficient (-0.0769230769). Square it (0.005917159760) and add it to both sides. Add '0.005917159760' to each side of the equation. -0.1538461538k + 0.005917159760 + k2 = 0.07692307692 + 0.005917159760 Reorder the terms: 0.005917159760 + -0.1538461538k + k2 = 0.07692307692 + 0.005917159760 Combine like terms: 0.07692307692 + 0.005917159760 = 0.08284023668 0.005917159760 + -0.1538461538k + k2 = 0.08284023668 Factor a perfect square on the left side: (k + -0.0769230769)(k + -0.0769230769) = 0.08284023668 Calculate the square root of the right side: 0.287819799 Break this problem into two subproblems by setting (k + -0.0769230769) equal to 0.287819799 and -0.287819799.Subproblem 1
k + -0.0769230769 = 0.287819799 Simplifying k + -0.0769230769 = 0.287819799 Reorder the terms: -0.0769230769 + k = 0.287819799 Solving -0.0769230769 + k = 0.287819799 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.0769230769' to each side of the equation. -0.0769230769 + 0.0769230769 + k = 0.287819799 + 0.0769230769 Combine like terms: -0.0769230769 + 0.0769230769 = 0.0000000000 0.0000000000 + k = 0.287819799 + 0.0769230769 k = 0.287819799 + 0.0769230769 Combine like terms: 0.287819799 + 0.0769230769 = 0.3647428759 k = 0.3647428759 Simplifying k = 0.3647428759Subproblem 2
k + -0.0769230769 = -0.287819799 Simplifying k + -0.0769230769 = -0.287819799 Reorder the terms: -0.0769230769 + k = -0.287819799 Solving -0.0769230769 + k = -0.287819799 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.0769230769' to each side of the equation. -0.0769230769 + 0.0769230769 + k = -0.287819799 + 0.0769230769 Combine like terms: -0.0769230769 + 0.0769230769 = 0.0000000000 0.0000000000 + k = -0.287819799 + 0.0769230769 k = -0.287819799 + 0.0769230769 Combine like terms: -0.287819799 + 0.0769230769 = -0.2108967221 k = -0.2108967221 Simplifying k = -0.2108967221Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.3647428759, -0.2108967221}
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