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Simplifying k2 + -4k + 1 = 0 Reorder the terms: 1 + -4k + k2 = 0 Solving 1 + -4k + 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 + -4k + -1 + k2 = 0 + -1 Reorder the terms: 1 + -1 + -4k + k2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -4k + k2 = 0 + -1 -4k + k2 = 0 + -1 Combine like terms: 0 + -1 = -1 -4k + k2 = -1 The k term is -4k. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4k + 4 + k2 = -1 + 4 Reorder the terms: 4 + -4k + k2 = -1 + 4 Combine like terms: -1 + 4 = 3 4 + -4k + k2 = 3 Factor a perfect square on the left side: (k + -2)(k + -2) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (k + -2) equal to 1.732050808 and -1.732050808.Subproblem 1
k + -2 = 1.732050808 Simplifying k + -2 = 1.732050808 Reorder the terms: -2 + k = 1.732050808 Solving -2 + k = 1.732050808 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + k = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + k = 1.732050808 + 2 k = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 k = 3.732050808 Simplifying k = 3.732050808Subproblem 2
k + -2 = -1.732050808 Simplifying k + -2 = -1.732050808 Reorder the terms: -2 + k = -1.732050808 Solving -2 + k = -1.732050808 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + k = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + k = -1.732050808 + 2 k = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 k = 0.267949192 Simplifying k = 0.267949192Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.732050808, 0.267949192}
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