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Simplifying 9k2 + k = 17 Reorder the terms: k + 9k2 = 17 Solving k + 9k2 = 17 Solving for variable 'k'. Reorder the terms: -17 + k + 9k2 = 17 + -17 Combine like terms: 17 + -17 = 0 -17 + k + 9k2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.888888889 + 0.1111111111k + k2 = 0 Move the constant term to the right: Add '1.888888889' to each side of the equation. -1.888888889 + 0.1111111111k + 1.888888889 + k2 = 0 + 1.888888889 Reorder the terms: -1.888888889 + 1.888888889 + 0.1111111111k + k2 = 0 + 1.888888889 Combine like terms: -1.888888889 + 1.888888889 = 0.000000000 0.000000000 + 0.1111111111k + k2 = 0 + 1.888888889 0.1111111111k + k2 = 0 + 1.888888889 Combine like terms: 0 + 1.888888889 = 1.888888889 0.1111111111k + k2 = 1.888888889 The k term is k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.1111111111k + 0.25 + k2 = 1.888888889 + 0.25 Reorder the terms: 0.25 + 0.1111111111k + k2 = 1.888888889 + 0.25 Combine like terms: 1.888888889 + 0.25 = 2.138888889 0.25 + 0.1111111111k + k2 = 2.138888889 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 2.138888889 Calculate the square root of the right side: 1.462494065 Break this problem into two subproblems by setting (k + 0.5) equal to 1.462494065 and -1.462494065.Subproblem 1
k + 0.5 = 1.462494065 Simplifying k + 0.5 = 1.462494065 Reorder the terms: 0.5 + k = 1.462494065 Solving 0.5 + k = 1.462494065 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 1.462494065 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 1.462494065 + -0.5 k = 1.462494065 + -0.5 Combine like terms: 1.462494065 + -0.5 = 0.962494065 k = 0.962494065 Simplifying k = 0.962494065Subproblem 2
k + 0.5 = -1.462494065 Simplifying k + 0.5 = -1.462494065 Reorder the terms: 0.5 + k = -1.462494065 Solving 0.5 + k = -1.462494065 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -1.462494065 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -1.462494065 + -0.5 k = -1.462494065 + -0.5 Combine like terms: -1.462494065 + -0.5 = -1.962494065 k = -1.962494065 Simplifying k = -1.962494065Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.962494065, -1.962494065}
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