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Simplifying k2 + -31 + -1k = -6 + -3k + -2k Reorder the terms: -31 + -1k + k2 = -6 + -3k + -2k Combine like terms: -3k + -2k = -5k -31 + -1k + k2 = -6 + -5k Solving -31 + -1k + k2 = -6 + -5k Solving for variable 'k'. Reorder the terms: -31 + 6 + -1k + 5k + k2 = -6 + -5k + 6 + 5k Combine like terms: -31 + 6 = -25 -25 + -1k + 5k + k2 = -6 + -5k + 6 + 5k Combine like terms: -1k + 5k = 4k -25 + 4k + k2 = -6 + -5k + 6 + 5k Reorder the terms: -25 + 4k + k2 = -6 + 6 + -5k + 5k Combine like terms: -6 + 6 = 0 -25 + 4k + k2 = 0 + -5k + 5k -25 + 4k + k2 = -5k + 5k Combine like terms: -5k + 5k = 0 -25 + 4k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '25' to each side of the equation. -25 + 4k + 25 + k2 = 0 + 25 Reorder the terms: -25 + 25 + 4k + k2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + 4k + k2 = 0 + 25 4k + k2 = 0 + 25 Combine like terms: 0 + 25 = 25 4k + k2 = 25 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 = 25 + 4 Reorder the terms: 4 + 4k + k2 = 25 + 4 Combine like terms: 25 + 4 = 29 4 + 4k + k2 = 29 Factor a perfect square on the left side: (k + 2)(k + 2) = 29 Calculate the square root of the right side: 5.385164807 Break this problem into two subproblems by setting (k + 2) equal to 5.385164807 and -5.385164807.Subproblem 1
k + 2 = 5.385164807 Simplifying k + 2 = 5.385164807 Reorder the terms: 2 + k = 5.385164807 Solving 2 + k = 5.385164807 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 = 5.385164807 + -2 Combine like terms: 2 + -2 = 0 0 + k = 5.385164807 + -2 k = 5.385164807 + -2 Combine like terms: 5.385164807 + -2 = 3.385164807 k = 3.385164807 Simplifying k = 3.385164807Subproblem 2
k + 2 = -5.385164807 Simplifying k + 2 = -5.385164807 Reorder the terms: 2 + k = -5.385164807 Solving 2 + k = -5.385164807 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 = -5.385164807 + -2 Combine like terms: 2 + -2 = 0 0 + k = -5.385164807 + -2 k = -5.385164807 + -2 Combine like terms: -5.385164807 + -2 = -7.385164807 k = -7.385164807 Simplifying k = -7.385164807Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.385164807, -7.385164807}
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