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Simplifying 2k2 = -17 + -12k Solving 2k2 = -17 + -12k Solving for variable 'k'. Reorder the terms: 17 + 12k + 2k2 = -17 + -12k + 17 + 12k Reorder the terms: 17 + 12k + 2k2 = -17 + 17 + -12k + 12k Combine like terms: -17 + 17 = 0 17 + 12k + 2k2 = 0 + -12k + 12k 17 + 12k + 2k2 = -12k + 12k Combine like terms: -12k + 12k = 0 17 + 12k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 8.5 + 6k + k2 = 0 Move the constant term to the right: Add '-8.5' to each side of the equation. 8.5 + 6k + -8.5 + k2 = 0 + -8.5 Reorder the terms: 8.5 + -8.5 + 6k + k2 = 0 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + 6k + k2 = 0 + -8.5 6k + k2 = 0 + -8.5 Combine like terms: 0 + -8.5 = -8.5 6k + k2 = -8.5 The k term is 6k. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6k + 9 + k2 = -8.5 + 9 Reorder the terms: 9 + 6k + k2 = -8.5 + 9 Combine like terms: -8.5 + 9 = 0.5 9 + 6k + k2 = 0.5 Factor a perfect square on the left side: (k + 3)(k + 3) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (k + 3) equal to 0.707106781 and -0.707106781.Subproblem 1
k + 3 = 0.707106781 Simplifying k + 3 = 0.707106781 Reorder the terms: 3 + k = 0.707106781 Solving 3 + k = 0.707106781 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + k = 0.707106781 + -3 Combine like terms: 3 + -3 = 0 0 + k = 0.707106781 + -3 k = 0.707106781 + -3 Combine like terms: 0.707106781 + -3 = -2.292893219 k = -2.292893219 Simplifying k = -2.292893219Subproblem 2
k + 3 = -0.707106781 Simplifying k + 3 = -0.707106781 Reorder the terms: 3 + k = -0.707106781 Solving 3 + k = -0.707106781 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + k = -0.707106781 + -3 Combine like terms: 3 + -3 = 0 0 + k = -0.707106781 + -3 k = -0.707106781 + -3 Combine like terms: -0.707106781 + -3 = -3.707106781 k = -3.707106781 Simplifying k = -3.707106781Solution
The solution to the problem is based on the solutions from the subproblems. k = {-2.292893219, -3.707106781}
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