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Simplifying k2 + 6k + 1 = 0 Reorder the terms: 1 + 6k + k2 = 0 Solving 1 + 6k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 6k + -1 + k2 = 0 + -1 Reorder the terms: 1 + -1 + 6k + k2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 6k + k2 = 0 + -1 6k + k2 = 0 + -1 Combine like terms: 0 + -1 = -1 6k + k2 = -1 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 = -1 + 9 Reorder the terms: 9 + 6k + k2 = -1 + 9 Combine like terms: -1 + 9 = 8 9 + 6k + k2 = 8 Factor a perfect square on the left side: (k + 3)(k + 3) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (k + 3) equal to 2.828427125 and -2.828427125.Subproblem 1
k + 3 = 2.828427125 Simplifying k + 3 = 2.828427125 Reorder the terms: 3 + k = 2.828427125 Solving 3 + k = 2.828427125 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 = 2.828427125 + -3 Combine like terms: 3 + -3 = 0 0 + k = 2.828427125 + -3 k = 2.828427125 + -3 Combine like terms: 2.828427125 + -3 = -0.171572875 k = -0.171572875 Simplifying k = -0.171572875Subproblem 2
k + 3 = -2.828427125 Simplifying k + 3 = -2.828427125 Reorder the terms: 3 + k = -2.828427125 Solving 3 + k = -2.828427125 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 = -2.828427125 + -3 Combine like terms: 3 + -3 = 0 0 + k = -2.828427125 + -3 k = -2.828427125 + -3 Combine like terms: -2.828427125 + -3 = -5.828427125 k = -5.828427125 Simplifying k = -5.828427125Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.171572875, -5.828427125}
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