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Simplifying 9k2 + 19k + -2 = 0 Reorder the terms: -2 + 19k + 9k2 = 0 Solving -2 + 19k + 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 + 2.111111111k + k2 = 0 Move the constant term to the right: Add '0.2222222222' to each side of the equation. -0.2222222222 + 2.111111111k + 0.2222222222 + k2 = 0 + 0.2222222222 Reorder the terms: -0.2222222222 + 0.2222222222 + 2.111111111k + k2 = 0 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + 2.111111111k + k2 = 0 + 0.2222222222 2.111111111k + k2 = 0 + 0.2222222222 Combine like terms: 0 + 0.2222222222 = 0.2222222222 2.111111111k + k2 = 0.2222222222 The k term is 2.111111111k. Take half its coefficient (1.055555556). Square it (1.114197532) and add it to both sides. Add '1.114197532' to each side of the equation. 2.111111111k + 1.114197532 + k2 = 0.2222222222 + 1.114197532 Reorder the terms: 1.114197532 + 2.111111111k + k2 = 0.2222222222 + 1.114197532 Combine like terms: 0.2222222222 + 1.114197532 = 1.3364197542 1.114197532 + 2.111111111k + k2 = 1.3364197542 Factor a perfect square on the left side: (k + 1.055555556)(k + 1.055555556) = 1.3364197542 Calculate the square root of the right side: 1.156036225 Break this problem into two subproblems by setting (k + 1.055555556) equal to 1.156036225 and -1.156036225.Subproblem 1
k + 1.055555556 = 1.156036225 Simplifying k + 1.055555556 = 1.156036225 Reorder the terms: 1.055555556 + k = 1.156036225 Solving 1.055555556 + k = 1.156036225 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.055555556' to each side of the equation. 1.055555556 + -1.055555556 + k = 1.156036225 + -1.055555556 Combine like terms: 1.055555556 + -1.055555556 = 0.000000000 0.000000000 + k = 1.156036225 + -1.055555556 k = 1.156036225 + -1.055555556 Combine like terms: 1.156036225 + -1.055555556 = 0.100480669 k = 0.100480669 Simplifying k = 0.100480669Subproblem 2
k + 1.055555556 = -1.156036225 Simplifying k + 1.055555556 = -1.156036225 Reorder the terms: 1.055555556 + k = -1.156036225 Solving 1.055555556 + k = -1.156036225 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.055555556' to each side of the equation. 1.055555556 + -1.055555556 + k = -1.156036225 + -1.055555556 Combine like terms: 1.055555556 + -1.055555556 = 0.000000000 0.000000000 + k = -1.156036225 + -1.055555556 k = -1.156036225 + -1.055555556 Combine like terms: -1.156036225 + -1.055555556 = -2.211591781 k = -2.211591781 Simplifying k = -2.211591781Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.100480669, -2.211591781}
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