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Simplifying -20k2 + 34k = 3k + -27 Reorder the terms: 34k + -20k2 = 3k + -27 Reorder the terms: 34k + -20k2 = -27 + 3k Solving 34k + -20k2 = -27 + 3k Solving for variable 'k'. Reorder the terms: 27 + 34k + -3k + -20k2 = -27 + 3k + 27 + -3k Combine like terms: 34k + -3k = 31k 27 + 31k + -20k2 = -27 + 3k + 27 + -3k Reorder the terms: 27 + 31k + -20k2 = -27 + 27 + 3k + -3k Combine like terms: -27 + 27 = 0 27 + 31k + -20k2 = 0 + 3k + -3k 27 + 31k + -20k2 = 3k + -3k Combine like terms: 3k + -3k = 0 27 + 31k + -20k2 = 0 Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. -1.35 + -1.55k + k2 = 0 Move the constant term to the right: Add '1.35' to each side of the equation. -1.35 + -1.55k + 1.35 + k2 = 0 + 1.35 Reorder the terms: -1.35 + 1.35 + -1.55k + k2 = 0 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + -1.55k + k2 = 0 + 1.35 -1.55k + k2 = 0 + 1.35 Combine like terms: 0 + 1.35 = 1.35 -1.55k + k2 = 1.35 The k term is -1.55k. Take half its coefficient (-0.775). Square it (0.600625) and add it to both sides. Add '0.600625' to each side of the equation. -1.55k + 0.600625 + k2 = 1.35 + 0.600625 Reorder the terms: 0.600625 + -1.55k + k2 = 1.35 + 0.600625 Combine like terms: 1.35 + 0.600625 = 1.950625 0.600625 + -1.55k + k2 = 1.950625 Factor a perfect square on the left side: (k + -0.775)(k + -0.775) = 1.950625 Calculate the square root of the right side: 1.396647772 Break this problem into two subproblems by setting (k + -0.775) equal to 1.396647772 and -1.396647772.Subproblem 1
k + -0.775 = 1.396647772 Simplifying k + -0.775 = 1.396647772 Reorder the terms: -0.775 + k = 1.396647772 Solving -0.775 + k = 1.396647772 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.775' to each side of the equation. -0.775 + 0.775 + k = 1.396647772 + 0.775 Combine like terms: -0.775 + 0.775 = 0.000 0.000 + k = 1.396647772 + 0.775 k = 1.396647772 + 0.775 Combine like terms: 1.396647772 + 0.775 = 2.171647772 k = 2.171647772 Simplifying k = 2.171647772Subproblem 2
k + -0.775 = -1.396647772 Simplifying k + -0.775 = -1.396647772 Reorder the terms: -0.775 + k = -1.396647772 Solving -0.775 + k = -1.396647772 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.775' to each side of the equation. -0.775 + 0.775 + k = -1.396647772 + 0.775 Combine like terms: -0.775 + 0.775 = 0.000 0.000 + k = -1.396647772 + 0.775 k = -1.396647772 + 0.775 Combine like terms: -1.396647772 + 0.775 = -0.621647772 k = -0.621647772 Simplifying k = -0.621647772Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.171647772, -0.621647772}
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