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Simplifying k2 + -18k = 54 Reorder the terms: -18k + k2 = 54 Solving -18k + k2 = 54 Solving for variable 'k'. Reorder the terms: -54 + -18k + k2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + -18k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '54' to each side of the equation. -54 + -18k + 54 + k2 = 0 + 54 Reorder the terms: -54 + 54 + -18k + k2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + -18k + k2 = 0 + 54 -18k + k2 = 0 + 54 Combine like terms: 0 + 54 = 54 -18k + k2 = 54 The k term is -18k. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18k + 81 + k2 = 54 + 81 Reorder the terms: 81 + -18k + k2 = 54 + 81 Combine like terms: 54 + 81 = 135 81 + -18k + k2 = 135 Factor a perfect square on the left side: (k + -9)(k + -9) = 135 Calculate the square root of the right side: 11.618950039 Break this problem into two subproblems by setting (k + -9) equal to 11.618950039 and -11.618950039.Subproblem 1
k + -9 = 11.618950039 Simplifying k + -9 = 11.618950039 Reorder the terms: -9 + k = 11.618950039 Solving -9 + k = 11.618950039 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + k = 11.618950039 + 9 Combine like terms: -9 + 9 = 0 0 + k = 11.618950039 + 9 k = 11.618950039 + 9 Combine like terms: 11.618950039 + 9 = 20.618950039 k = 20.618950039 Simplifying k = 20.618950039Subproblem 2
k + -9 = -11.618950039 Simplifying k + -9 = -11.618950039 Reorder the terms: -9 + k = -11.618950039 Solving -9 + k = -11.618950039 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + k = -11.618950039 + 9 Combine like terms: -9 + 9 = 0 0 + k = -11.618950039 + 9 k = -11.618950039 + 9 Combine like terms: -11.618950039 + 9 = -2.618950039 k = -2.618950039 Simplifying k = -2.618950039Solution
The solution to the problem is based on the solutions from the subproblems. k = {20.618950039, -2.618950039}
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