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Simplifying k * k + -6k + 1 = 0 Multiply k * k 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 = 5.828427125 k = 5.828427125 Simplifying k = 5.828427125Subproblem 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 = 0.171572875 k = 0.171572875 Simplifying k = 0.171572875Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.828427125, 0.171572875}
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