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Simplifying (8k3 + 5k4 + -11k5) + (-6k5 + -13k4 + -3k3) = 0 Remove parenthesis around (8k3 + 5k4 + -11k5) 8k3 + 5k4 + -11k5 + (-6k5 + -13k4 + -3k3) = 0 Reorder the terms: 8k3 + 5k4 + -11k5 + (-3k3 + -13k4 + -6k5) = 0 Remove parenthesis around (-3k3 + -13k4 + -6k5) 8k3 + 5k4 + -11k5 + -3k3 + -13k4 + -6k5 = 0 Reorder the terms: 8k3 + -3k3 + 5k4 + -13k4 + -11k5 + -6k5 = 0 Combine like terms: 8k3 + -3k3 = 5k3 5k3 + 5k4 + -13k4 + -11k5 + -6k5 = 0 Combine like terms: 5k4 + -13k4 = -8k4 5k3 + -8k4 + -11k5 + -6k5 = 0 Combine like terms: -11k5 + -6k5 = -17k5 5k3 + -8k4 + -17k5 = 0 Solving 5k3 + -8k4 + -17k5 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), 'k3'. k3(5 + -8k + -17k2) = 0Subproblem 1
Set the factor 'k3' equal to zero and attempt to solve: Simplifying k3 = 0 Solving k3 = 0 Move all terms containing k to the left, all other terms to the right. Simplifying k3 = 0 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(5 + -8k + -17k2)' equal to zero and attempt to solve: Simplifying 5 + -8k + -17k2 = 0 Solving 5 + -8k + -17k2 = 0 Begin completing the square. Divide all terms by -17 the coefficient of the squared term: Divide each side by '-17'. -0.2941176471 + 0.4705882353k + k2 = 0 Move the constant term to the right: Add '0.2941176471' to each side of the equation. -0.2941176471 + 0.4705882353k + 0.2941176471 + k2 = 0 + 0.2941176471 Reorder the terms: -0.2941176471 + 0.2941176471 + 0.4705882353k + k2 = 0 + 0.2941176471 Combine like terms: -0.2941176471 + 0.2941176471 = 0.0000000000 0.0000000000 + 0.4705882353k + k2 = 0 + 0.2941176471 0.4705882353k + k2 = 0 + 0.2941176471 Combine like terms: 0 + 0.2941176471 = 0.2941176471 0.4705882353k + k2 = 0.2941176471 The k term is 0.4705882353k. Take half its coefficient (0.2352941177). Square it (0.05536332182) and add it to both sides. Add '0.05536332182' to each side of the equation. 0.4705882353k + 0.05536332182 + k2 = 0.2941176471 + 0.05536332182 Reorder the terms: 0.05536332182 + 0.4705882353k + k2 = 0.2941176471 + 0.05536332182 Combine like terms: 0.2941176471 + 0.05536332182 = 0.34948096892 0.05536332182 + 0.4705882353k + k2 = 0.34948096892 Factor a perfect square on the left side: (k + 0.2352941177)(k + 0.2352941177) = 0.34948096892 Calculate the square root of the right side: 0.591169154 Break this problem into two subproblems by setting (k + 0.2352941177) equal to 0.591169154 and -0.591169154.Subproblem 1
k + 0.2352941177 = 0.591169154 Simplifying k + 0.2352941177 = 0.591169154 Reorder the terms: 0.2352941177 + k = 0.591169154 Solving 0.2352941177 + k = 0.591169154 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.2352941177' to each side of the equation. 0.2352941177 + -0.2352941177 + k = 0.591169154 + -0.2352941177 Combine like terms: 0.2352941177 + -0.2352941177 = 0.0000000000 0.0000000000 + k = 0.591169154 + -0.2352941177 k = 0.591169154 + -0.2352941177 Combine like terms: 0.591169154 + -0.2352941177 = 0.3558750363 k = 0.3558750363 Simplifying k = 0.3558750363Subproblem 2
k + 0.2352941177 = -0.591169154 Simplifying k + 0.2352941177 = -0.591169154 Reorder the terms: 0.2352941177 + k = -0.591169154 Solving 0.2352941177 + k = -0.591169154 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.2352941177' to each side of the equation. 0.2352941177 + -0.2352941177 + k = -0.591169154 + -0.2352941177 Combine like terms: 0.2352941177 + -0.2352941177 = 0.0000000000 0.0000000000 + k = -0.591169154 + -0.2352941177 k = -0.591169154 + -0.2352941177 Combine like terms: -0.591169154 + -0.2352941177 = -0.8264632717 k = -0.8264632717 Simplifying k = -0.8264632717Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.3558750363, -0.8264632717}Solution
k = {0.3558750363, -0.8264632717}
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