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Simplifying 27k = k2 + -9 Reorder the terms: 27k = -9 + k2 Solving 27k = -9 + k2 Solving for variable 'k'. Reorder the terms: 9 + 27k + -1k2 = -9 + k2 + 9 + -1k2 Reorder the terms: 9 + 27k + -1k2 = -9 + 9 + k2 + -1k2 Combine like terms: -9 + 9 = 0 9 + 27k + -1k2 = 0 + k2 + -1k2 9 + 27k + -1k2 = k2 + -1k2 Combine like terms: k2 + -1k2 = 0 9 + 27k + -1k2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -9 + -27k + k2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -27k + 9 + k2 = 0 + 9 Reorder the terms: -9 + 9 + -27k + k2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -27k + k2 = 0 + 9 -27k + k2 = 0 + 9 Combine like terms: 0 + 9 = 9 -27k + k2 = 9 The k term is -27k. Take half its coefficient (-13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. -27k + 182.25 + k2 = 9 + 182.25 Reorder the terms: 182.25 + -27k + k2 = 9 + 182.25 Combine like terms: 9 + 182.25 = 191.25 182.25 + -27k + k2 = 191.25 Factor a perfect square on the left side: (k + -13.5)(k + -13.5) = 191.25 Calculate the square root of the right side: 13.829316686 Break this problem into two subproblems by setting (k + -13.5) equal to 13.829316686 and -13.829316686.Subproblem 1
k + -13.5 = 13.829316686 Simplifying k + -13.5 = 13.829316686 Reorder the terms: -13.5 + k = 13.829316686 Solving -13.5 + k = 13.829316686 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + k = 13.829316686 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + k = 13.829316686 + 13.5 k = 13.829316686 + 13.5 Combine like terms: 13.829316686 + 13.5 = 27.329316686 k = 27.329316686 Simplifying k = 27.329316686Subproblem 2
k + -13.5 = -13.829316686 Simplifying k + -13.5 = -13.829316686 Reorder the terms: -13.5 + k = -13.829316686 Solving -13.5 + k = -13.829316686 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + k = -13.829316686 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + k = -13.829316686 + 13.5 k = -13.829316686 + 13.5 Combine like terms: -13.829316686 + 13.5 = -0.329316686 k = -0.329316686 Simplifying k = -0.329316686Solution
The solution to the problem is based on the solutions from the subproblems. k = {27.329316686, -0.329316686}
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