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Simplifying k2 + -12k + 23 = 0e Reorder the terms: 23 + -12k + k2 = 0e Anything times zero is zero. 23 + -12k + k2 = 0e Solving 23 + -12k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '-23' to each side of the equation. 23 + -12k + -23 + k2 = 0 + -23 Reorder the terms: 23 + -23 + -12k + k2 = 0 + -23 Combine like terms: 23 + -23 = 0 0 + -12k + k2 = 0 + -23 -12k + k2 = 0 + -23 Combine like terms: 0 + -23 = -23 -12k + k2 = -23 The k term is -12k. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12k + 36 + k2 = -23 + 36 Reorder the terms: 36 + -12k + k2 = -23 + 36 Combine like terms: -23 + 36 = 13 36 + -12k + k2 = 13 Factor a perfect square on the left side: (k + -6)(k + -6) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (k + -6) equal to 3.605551275 and -3.605551275.Subproblem 1
k + -6 = 3.605551275 Simplifying k + -6 = 3.605551275 Reorder the terms: -6 + k = 3.605551275 Solving -6 + k = 3.605551275 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + k = 3.605551275 + 6 Combine like terms: -6 + 6 = 0 0 + k = 3.605551275 + 6 k = 3.605551275 + 6 Combine like terms: 3.605551275 + 6 = 9.605551275 k = 9.605551275 Simplifying k = 9.605551275Subproblem 2
k + -6 = -3.605551275 Simplifying k + -6 = -3.605551275 Reorder the terms: -6 + k = -3.605551275 Solving -6 + k = -3.605551275 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + k = -3.605551275 + 6 Combine like terms: -6 + 6 = 0 0 + k = -3.605551275 + 6 k = -3.605551275 + 6 Combine like terms: -3.605551275 + 6 = 2.394448725 k = 2.394448725 Simplifying k = 2.394448725Solution
The solution to the problem is based on the solutions from the subproblems. k = {9.605551275, 2.394448725}
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