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Simplifying k2 + 14k = -46 Reorder the terms: 14k + k2 = -46 Solving 14k + k2 = -46 Solving for variable 'k'. Reorder the terms: 46 + 14k + k2 = -46 + 46 Combine like terms: -46 + 46 = 0 46 + 14k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '-46' to each side of the equation. 46 + 14k + -46 + k2 = 0 + -46 Reorder the terms: 46 + -46 + 14k + k2 = 0 + -46 Combine like terms: 46 + -46 = 0 0 + 14k + k2 = 0 + -46 14k + k2 = 0 + -46 Combine like terms: 0 + -46 = -46 14k + k2 = -46 The k term is 14k. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14k + 49 + k2 = -46 + 49 Reorder the terms: 49 + 14k + k2 = -46 + 49 Combine like terms: -46 + 49 = 3 49 + 14k + k2 = 3 Factor a perfect square on the left side: (k + 7)(k + 7) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (k + 7) equal to 1.732050808 and -1.732050808.Subproblem 1
k + 7 = 1.732050808 Simplifying k + 7 = 1.732050808 Reorder the terms: 7 + k = 1.732050808 Solving 7 + k = 1.732050808 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + k = 1.732050808 + -7 Combine like terms: 7 + -7 = 0 0 + k = 1.732050808 + -7 k = 1.732050808 + -7 Combine like terms: 1.732050808 + -7 = -5.267949192 k = -5.267949192 Simplifying k = -5.267949192Subproblem 2
k + 7 = -1.732050808 Simplifying k + 7 = -1.732050808 Reorder the terms: 7 + k = -1.732050808 Solving 7 + k = -1.732050808 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + k = -1.732050808 + -7 Combine like terms: 7 + -7 = 0 0 + k = -1.732050808 + -7 k = -1.732050808 + -7 Combine like terms: -1.732050808 + -7 = -8.732050808 k = -8.732050808 Simplifying k = -8.732050808Solution
The solution to the problem is based on the solutions from the subproblems. k = {-5.267949192, -8.732050808}
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