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Simplifying k2 + 2k + -86 = -9 Reorder the terms: -86 + 2k + k2 = -9 Solving -86 + 2k + k2 = -9 Solving for variable 'k'. Reorder the terms: -86 + 9 + 2k + k2 = -9 + 9 Combine like terms: -86 + 9 = -77 -77 + 2k + k2 = -9 + 9 Combine like terms: -9 + 9 = 0 -77 + 2k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '77' to each side of the equation. -77 + 2k + 77 + k2 = 0 + 77 Reorder the terms: -77 + 77 + 2k + k2 = 0 + 77 Combine like terms: -77 + 77 = 0 0 + 2k + k2 = 0 + 77 2k + k2 = 0 + 77 Combine like terms: 0 + 77 = 77 2k + k2 = 77 The k term is 2k. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2k + 1 + k2 = 77 + 1 Reorder the terms: 1 + 2k + k2 = 77 + 1 Combine like terms: 77 + 1 = 78 1 + 2k + k2 = 78 Factor a perfect square on the left side: (k + 1)(k + 1) = 78 Calculate the square root of the right side: 8.831760866 Break this problem into two subproblems by setting (k + 1) equal to 8.831760866 and -8.831760866.Subproblem 1
k + 1 = 8.831760866 Simplifying k + 1 = 8.831760866 Reorder the terms: 1 + k = 8.831760866 Solving 1 + k = 8.831760866 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = 8.831760866 + -1 Combine like terms: 1 + -1 = 0 0 + k = 8.831760866 + -1 k = 8.831760866 + -1 Combine like terms: 8.831760866 + -1 = 7.831760866 k = 7.831760866 Simplifying k = 7.831760866Subproblem 2
k + 1 = -8.831760866 Simplifying k + 1 = -8.831760866 Reorder the terms: 1 + k = -8.831760866 Solving 1 + k = -8.831760866 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = -8.831760866 + -1 Combine like terms: 1 + -1 = 0 0 + k = -8.831760866 + -1 k = -8.831760866 + -1 Combine like terms: -8.831760866 + -1 = -9.831760866 k = -9.831760866 Simplifying k = -9.831760866Solution
The solution to the problem is based on the solutions from the subproblems. k = {7.831760866, -9.831760866}
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