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Simplifying 0 = 2k2 + -12k + 5 Reorder the terms: 0 = 5 + -12k + 2k2 Solving 0 = 5 + -12k + 2k2 Solving for variable 'k'. Combine like terms: 0 + -5 = -5 -5 + 12k + -2k2 = 5 + -12k + 2k2 + -5 + 12k + -2k2 Reorder the terms: -5 + 12k + -2k2 = 5 + -5 + -12k + 12k + 2k2 + -2k2 Combine like terms: 5 + -5 = 0 -5 + 12k + -2k2 = 0 + -12k + 12k + 2k2 + -2k2 -5 + 12k + -2k2 = -12k + 12k + 2k2 + -2k2 Combine like terms: -12k + 12k = 0 -5 + 12k + -2k2 = 0 + 2k2 + -2k2 -5 + 12k + -2k2 = 2k2 + -2k2 Combine like terms: 2k2 + -2k2 = 0 -5 + 12k + -2k2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 2.5 + -6k + k2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -6k + -2.5 + k2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -6k + k2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -6k + k2 = 0 + -2.5 -6k + k2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -6k + k2 = -2.5 The k term is -6k. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6k + 9 + k2 = -2.5 + 9 Reorder the terms: 9 + -6k + k2 = -2.5 + 9 Combine like terms: -2.5 + 9 = 6.5 9 + -6k + k2 = 6.5 Factor a perfect square on the left side: (k + -3)(k + -3) = 6.5 Calculate the square root of the right side: 2.549509757 Break this problem into two subproblems by setting (k + -3) equal to 2.549509757 and -2.549509757.Subproblem 1
k + -3 = 2.549509757 Simplifying k + -3 = 2.549509757 Reorder the terms: -3 + k = 2.549509757 Solving -3 + k = 2.549509757 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + k = 2.549509757 + 3 Combine like terms: -3 + 3 = 0 0 + k = 2.549509757 + 3 k = 2.549509757 + 3 Combine like terms: 2.549509757 + 3 = 5.549509757 k = 5.549509757 Simplifying k = 5.549509757Subproblem 2
k + -3 = -2.549509757 Simplifying k + -3 = -2.549509757 Reorder the terms: -3 + k = -2.549509757 Solving -3 + k = -2.549509757 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + k = -2.549509757 + 3 Combine like terms: -3 + 3 = 0 0 + k = -2.549509757 + 3 k = -2.549509757 + 3 Combine like terms: -2.549509757 + 3 = 0.450490243 k = 0.450490243 Simplifying k = 0.450490243Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.549509757, 0.450490243}
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