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Simplifying 9k2 + 56k + -2 = 0 Reorder the terms: -2 + 56k + 9k2 = 0 Solving -2 + 56k + 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'. -0.2222222222 + 6.222222222k + k2 = 0 Move the constant term to the right: Add '0.2222222222' to each side of the equation. -0.2222222222 + 6.222222222k + 0.2222222222 + k2 = 0 + 0.2222222222 Reorder the terms: -0.2222222222 + 0.2222222222 + 6.222222222k + k2 = 0 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + 6.222222222k + k2 = 0 + 0.2222222222 6.222222222k + k2 = 0 + 0.2222222222 Combine like terms: 0 + 0.2222222222 = 0.2222222222 6.222222222k + k2 = 0.2222222222 The k term is 6.222222222k. Take half its coefficient (3.111111111). Square it (9.679012345) and add it to both sides. Add '9.679012345' to each side of the equation. 6.222222222k + 9.679012345 + k2 = 0.2222222222 + 9.679012345 Reorder the terms: 9.679012345 + 6.222222222k + k2 = 0.2222222222 + 9.679012345 Combine like terms: 0.2222222222 + 9.679012345 = 9.9012345672 9.679012345 + 6.222222222k + k2 = 9.9012345672 Factor a perfect square on the left side: (k + 3.111111111)(k + 3.111111111) = 9.9012345672 Calculate the square root of the right side: 3.146622724 Break this problem into two subproblems by setting (k + 3.111111111) equal to 3.146622724 and -3.146622724.Subproblem 1
k + 3.111111111 = 3.146622724 Simplifying k + 3.111111111 = 3.146622724 Reorder the terms: 3.111111111 + k = 3.146622724 Solving 3.111111111 + k = 3.146622724 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.111111111' to each side of the equation. 3.111111111 + -3.111111111 + k = 3.146622724 + -3.111111111 Combine like terms: 3.111111111 + -3.111111111 = 0.000000000 0.000000000 + k = 3.146622724 + -3.111111111 k = 3.146622724 + -3.111111111 Combine like terms: 3.146622724 + -3.111111111 = 0.035511613 k = 0.035511613 Simplifying k = 0.035511613Subproblem 2
k + 3.111111111 = -3.146622724 Simplifying k + 3.111111111 = -3.146622724 Reorder the terms: 3.111111111 + k = -3.146622724 Solving 3.111111111 + k = -3.146622724 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.111111111' to each side of the equation. 3.111111111 + -3.111111111 + k = -3.146622724 + -3.111111111 Combine like terms: 3.111111111 + -3.111111111 = 0.000000000 0.000000000 + k = -3.146622724 + -3.111111111 k = -3.146622724 + -3.111111111 Combine like terms: -3.146622724 + -3.111111111 = -6.257733835 k = -6.257733835 Simplifying k = -6.257733835Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.035511613, -6.257733835}
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