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Simplifying 9k2 + -1k + -1 = 0 Reorder the terms: -1 + -1k + 9k2 = 0 Solving -1 + -1k + 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'. -0.1111111111 + -0.1111111111k + k2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -0.1111111111k + 0.1111111111 + k2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -0.1111111111k + k2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -0.1111111111k + k2 = 0 + 0.1111111111 -0.1111111111k + k2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -0.1111111111k + k2 = 0.1111111111 The k term is -0.1111111111k. Take half its coefficient (-0.05555555555). Square it (0.003086419752) and add it to both sides. Add '0.003086419752' to each side of the equation. -0.1111111111k + 0.003086419752 + k2 = 0.1111111111 + 0.003086419752 Reorder the terms: 0.003086419752 + -0.1111111111k + k2 = 0.1111111111 + 0.003086419752 Combine like terms: 0.1111111111 + 0.003086419752 = 0.114197530852 0.003086419752 + -0.1111111111k + k2 = 0.114197530852 Factor a perfect square on the left side: (k + -0.05555555555)(k + -0.05555555555) = 0.114197530852 Calculate the square root of the right side: 0.337931252 Break this problem into two subproblems by setting (k + -0.05555555555) equal to 0.337931252 and -0.337931252.Subproblem 1
k + -0.05555555555 = 0.337931252 Simplifying k + -0.05555555555 = 0.337931252 Reorder the terms: -0.05555555555 + k = 0.337931252 Solving -0.05555555555 + k = 0.337931252 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.05555555555' to each side of the equation. -0.05555555555 + 0.05555555555 + k = 0.337931252 + 0.05555555555 Combine like terms: -0.05555555555 + 0.05555555555 = 0.00000000000 0.00000000000 + k = 0.337931252 + 0.05555555555 k = 0.337931252 + 0.05555555555 Combine like terms: 0.337931252 + 0.05555555555 = 0.39348680755 k = 0.39348680755 Simplifying k = 0.39348680755Subproblem 2
k + -0.05555555555 = -0.337931252 Simplifying k + -0.05555555555 = -0.337931252 Reorder the terms: -0.05555555555 + k = -0.337931252 Solving -0.05555555555 + k = -0.337931252 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.05555555555' to each side of the equation. -0.05555555555 + 0.05555555555 + k = -0.337931252 + 0.05555555555 Combine like terms: -0.05555555555 + 0.05555555555 = 0.00000000000 0.00000000000 + k = -0.337931252 + 0.05555555555 k = -0.337931252 + 0.05555555555 Combine like terms: -0.337931252 + 0.05555555555 = -0.28237569645 k = -0.28237569645 Simplifying k = -0.28237569645Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.39348680755, -0.28237569645}
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