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Simplifying 9k2 + 29k + 8 = 0 Reorder the terms: 8 + 29k + 9k2 = 0 Solving 8 + 29k + 9k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.8888888889 + 3.222222222k + k2 = 0 Move the constant term to the right: Add '-0.8888888889' to each side of the equation. 0.8888888889 + 3.222222222k + -0.8888888889 + k2 = 0 + -0.8888888889 Reorder the terms: 0.8888888889 + -0.8888888889 + 3.222222222k + k2 = 0 + -0.8888888889 Combine like terms: 0.8888888889 + -0.8888888889 = 0.0000000000 0.0000000000 + 3.222222222k + k2 = 0 + -0.8888888889 3.222222222k + k2 = 0 + -0.8888888889 Combine like terms: 0 + -0.8888888889 = -0.8888888889 3.222222222k + k2 = -0.8888888889 The k term is 3.222222222k. Take half its coefficient (1.611111111). Square it (2.595679012) and add it to both sides. Add '2.595679012' to each side of the equation. 3.222222222k + 2.595679012 + k2 = -0.8888888889 + 2.595679012 Reorder the terms: 2.595679012 + 3.222222222k + k2 = -0.8888888889 + 2.595679012 Combine like terms: -0.8888888889 + 2.595679012 = 1.7067901231 2.595679012 + 3.222222222k + k2 = 1.7067901231 Factor a perfect square on the left side: (k + 1.611111111)(k + 1.611111111) = 1.7067901231 Calculate the square root of the right side: 1.306441779 Break this problem into two subproblems by setting (k + 1.611111111) equal to 1.306441779 and -1.306441779.Subproblem 1
k + 1.611111111 = 1.306441779 Simplifying k + 1.611111111 = 1.306441779 Reorder the terms: 1.611111111 + k = 1.306441779 Solving 1.611111111 + k = 1.306441779 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.611111111' to each side of the equation. 1.611111111 + -1.611111111 + k = 1.306441779 + -1.611111111 Combine like terms: 1.611111111 + -1.611111111 = 0.000000000 0.000000000 + k = 1.306441779 + -1.611111111 k = 1.306441779 + -1.611111111 Combine like terms: 1.306441779 + -1.611111111 = -0.304669332 k = -0.304669332 Simplifying k = -0.304669332Subproblem 2
k + 1.611111111 = -1.306441779 Simplifying k + 1.611111111 = -1.306441779 Reorder the terms: 1.611111111 + k = -1.306441779 Solving 1.611111111 + k = -1.306441779 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.611111111' to each side of the equation. 1.611111111 + -1.611111111 + k = -1.306441779 + -1.611111111 Combine like terms: 1.611111111 + -1.611111111 = 0.000000000 0.000000000 + k = -1.306441779 + -1.611111111 k = -1.306441779 + -1.611111111 Combine like terms: -1.306441779 + -1.611111111 = -2.91755289 k = -2.91755289 Simplifying k = -2.91755289Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.304669332, -2.91755289}
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